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U. S. Naval Postgraduate School 

Monterey, California 



THE LIFT COEFFICIMT OF FLAT PLANING SURFACES 



A Thesis Submitted in Partial Fulfillment 

of the Requirements for the Degree of 

Master of Science 



U. S. Naval Postgraduate Schqol 
STE7MS INSTITUTE OF TECHNOLOGY Njonterey. Cnlifornia 



Submitted by Donald David Farshing, Jr. 



TABLE OF CONTENTS 



Page 



Abstract 

Notation 

Introduction 1 

Modification of the Two-Dim ens ional Planing Solution for 

the Effect of a Finite Span in a Wfeightless Fluid 3 

Historical Background 3 

The Two -Dimensional Solution 3 

Application of the Momentum Theorem - Method I 6 

Application of the Momentum Theorem - Method II III 

Experimental Investigation 16 

Purpose 16 

Equipment and Procedure 16 

EjqDerimental Results 21 

Comparison of Theoretical Results with Experimental Data 22 

General Discussion 22 

Empirical Correction to the Curves of C, Computed by 

the Cubic Equation 22 

The Effect of Gravity 2ii 

Summary and Conclusions 2? 

Bibliography 31 

Appendix I 3^ 

Appendix II h^ 

Tables ^k 

Figures 



^S'7'5'^ 



ABSTRACT 

A theoretical and experimental investigation bo determine the 
lift coefficient of a flat plate planing surface covering a broad 
range of trim angles and aspect ratios is presented. 

The analysis is divided into two principal parts: first con- 
sidered is the modification of the two-dimensional potential flow 
solution for the effect of a finite span neglecting gravity; then, 
second, the effects of gravity on the finite span planing surface 
are discussed with the importance of Froude Number emphasized. 

The analysis made on the effect of a finite span neglecting 
gravity is accompanied by an experimental investigation in which 
angles of trim up to 30 degrees and wetted length -beam ratios up to 
about seven are explored. 

The analytical approach leads to the prediction of reasonable 
values for the lift coefficient, but to get better agreement with 
experimental data, an empirical correction factor is introduced. 

The investigation, conducted at the Experimental Towing Tank, 
Stevens institute of Technology, Hoboken, N. J., is the thesis sub- 
mitted by the author in partial fulfillment of the requirements for 
the degree of Master of Science. 



NOTATION 

TT Geometric angle of trim 

X Wetted length-beam ratio (— ) 

b 

C^ Speed coefficient (Froude Number) V/Vgb 

g Acceleration of gravity 

b Beam of planing surface or span of airfoil 

V Velocity of the free stream 

C^ Lift coefficient, r — 

^ ipV^bi 

L Lift force 

r Fluid density 

jZ Wetted length of planing surface or chord of airfoil 

m 

S Projected wing area, I I cfydx 

F Normal force ** 



1 b^ 
Aspect ratio, r- or ^ 



■ If- 



0( Ambient flow angle of attack 

q/. Induced angle of attack 

O Spray thickness 

V. Induced downwash velocity infinitely far aft 

lac 

m Virtual mass rate of flow, VpVol. 

KE Kinetic energy 

/I^KE Change in kinetic energy 

r Wetted length measured from trailing edge to stagnation 
line on planing surface 

A O.ht^ 

B '^^ h sinT 



A 

^~- Einpirical wave rise factor 

r-^ Wetted length-beam ratio neglecting wave rise 

Vi/Vz Snpirical correction factor 



Notation Used in the Section on the Effect of Gravity 

Velocity potential 

U Free stream velocity 

5, '7 Coordinates of point (x,y) on a planing surface 

(x,y,z) Coordinates of a point in space 

10 Wave number ^p 

U 

IjL Coefficient introduced by Maruo to obtain desired value 

^ of certain integrals 

P (f,"*7) Pressure at any point on planing surface 

9 Wave direction angle 

k General wave number 

w Downwash as used by Maruo 

1^ A function of Froude Number 

B Buoyancy force (Archi median Force) 

Notation in Appendix 1 

^ Geometric angle of attack 

oC Ambient flow angle of attack 
e 

w^ Induced angle of attack 

w Complex potential 

Velocity potential 

Y stream function 

z Complex variable, z = x + iy 



T Kinetic energy 

S Surface 

V Transverse flow velocity 

Notation in Appendix II 

R Force 

R Drag force 

X 

Q Volume flow in spray jet, Q = V^ 

06 Angle of trim 

I^ Spray resistance 

D Total lift f orce, dynamic plus Archiniedian 

R^ Frictional Force 
tp 

R^ Wave resistance 



V A function of Froude Number^ = r- 

2 2 Fr 

2 V 

Fr Froude Number squared 5 —q 

M Moment 

C Lift coefficient (C ) 

•'''d Center of pressure location from trailing edge 

P, S, R Vector forces 

h Depth of fluid from free surface 



THE LIFT COEFFICIENT OF FLAT PLAINING SURFACES 

TOTR.nTITnTTnN 

Ifydrodynamic phenomena arising from planing on the surface of 
water have aroused considerable interest in seaplane hull^ float, and 
hydroski enthusiasts. In the past the need for immediate design data 
for floats or hydroskis has led, based on many specific experimental 
investigations, to the development of numerous empirical formulae to 
describe the flat plate lift coefficient as it varies with other para- 
meters such as angle of trim^ 'V, wetted length-beam ratio, X, and 
Froude Number, V/vgb, While these empirical formulae agree very well 
with experimental results within the range of values for which they 
were developed^ the validity of such formulae outside the range of data 
for which they were developed becomes unreliable or at least question- 
able. 

The objective of this work is to develop on a rational basis, 
insofar as possible, a fonmila for the lift coefficient of a flat 
plate which planes on the surface of water. At present, a rationally 
derived formula developed by H„ Wagner, Ref . 1, covers the infinite 
span case, neglecting gravity, while the more recent work of L„ I. Sedov, 
Ref, 2, and H, Maruo, Ref, 3, covers the infinite span including the 
effect of fluid gravity which involves wavemaking as well as hydrostatic 



effects. Methods of correction for the effect of a finite span neg- 
lecting gravity have been developed by such writers as B. V. Korvin- 
Kroukovsky, Byrne Perry, W. Bollay, W. G. A. Perring and L. Johnston, 
W. Sottorfy L. Landweber, F. W. S. Locke, and G. L. Shuford, Jr., 
Ref. ii-12, but the entire process, including the effects of trim, 
aspect ratio, and Froude Nximber, has not been integrated except in 
a preliminary form. 

This work involves an investigation of the methods and applic- 
ability of derivations previously made by the investigators of flat- 
plate planiQg. The problem of deriving a formula on a rational basis 
is treated in two distinct parts. First, the aspect ratio, trim angle 
effect^ neglecting gravity, is considered; then, second, the effects of 
gravity including wavemaking are discussed. 

To extend the range of experimental data for comparison with re- 
sults developed analytically, an experimental investigation was made 
which is described herein. 

The investigation has been conducted by Lto D., D. Farshing, Jr., 
U.SoN., a Naval Postgraduate School student at the Experimental Towing 
Tank, Stevens Institute of Technology, Hoboken, New Jersey. Acknow- 
ledgement is made to Professor B. V. Korvin-Kroukovsky of the Exper- 
imental Towing Tank for his helpful guidance and interest in this work. 
Gratitude is also expressed to the members of the Towing Tank staff Yiho 
have willingly given their assistance and advice throughout the course 
of this investigation. 



MODIFICATION OF THE TWO-DIMENSIONAL PLANING SOLUTION 
FOR THE EFFECT OF A FINITE SPAN IN A WEIGHTLESS FLUID 

Historical Background 

There have been several attempts to find closed form solutions 
to the finite span planing surface problan, but difficulty arises in 
that formulation of the problem to satisfy the boundary conditions 
introduces a non-linear integral equation. Solution of the integral 
equation is feasible only after many simplifying assumptions have been 
made. 

To circumvent the difficulty arising from the integral equation, 
another approach is offered. An ideal weightless fluid is assimied to 
exist. Based on the method of potential flow, a two-dimensional solu- 
tion is found, then modified for the effect of a finite span by a tech- 
nique similar to that used for correcting the two-dimensional airfoil 
to finite spano 

A brief but comprehensive review of the significant work in plan- 
ing up to 19h9 has been presented by William Siler, Ref . 13 • Charles 
L. Shuford, Jr., Ref« 12, has given a recapitulation of the results of 
many investigators who sought to find a simple formula for the three- 
dimensional lift coefficient of the planing flat plate. After his re- 
view of the attempts of other investigators to find simple formulas, 
Shuford proposes another method for predicting the lift coefficient. 
Shuford observes that experimental data indicates a non-linear rela- 
tionship between the angle of trim and the planing lift coefficient 



for the planing surface of finite span. To account for the non- 
linearity, he approaches the problem with an analogy to the low aspect 
ratio airfoil. Appendix II shows further justification for this ap- 
proach. To arrive at a final total lift formula, Shuford combines 
three components; the first, a linear term given as one -half the value 
of the flat plate airfoil lift coefficient based on the lifting line 

theory; the second, a non-linear, cross flow term which is shown to be 

2 
proportional to sin ^f', and the thirds a suction component of lift 

existing at the leading edge of the airfoil but not present in the 

planing surface flow. 

Shuford' s total lift formula becomes 

„ r/2 mr ,. . 2 X „ 2_ 

L ^ 1-^PB. ~ • ~ ^"^ ^' * ^^ ^°^ '^ 
values of which agree well with experimental data„ The analysis by 
Shuford, while in good agreement with the available experimental data, 
is not completely theoretical since the planing problem was not solved, 
but, instead, the airfoil problem manipulated. 

In many of the papers concerned with modification of the two- 
dimensional solution for the effect of finite span, methods similar to 
those used for correcting the infinite span airfoil have been used with 
varying degrees of success „ The technique of adopting similar methods 
seems to be a logical approach since the flow is assumed ideal (without 
friction) and weightless. 

Another successful approach is one made by Byrne Perry, Ref . 6. 
Perry has applied the low aspect ratio airfoil analysis of Fo Weinig, 
Ref o lii, to the low aspect ratio planing surface. Weinig solved the 



airfoil problem without recourse to the two-dimensional solution. He 
applied the momentum theorem to a control surface around the airfoil 
with virtual mass and induced angle of attack as independent variables. 
The virtual mass used by Weinig is that associated with the lifting 
line plus a wedge of fluid found by the projection of wetted length 
normal to the flow: 

J sin tr. (See Fig. 1) 
Appendix I presents more in detail the concept of virtual mass and the 
work of Weinig insofar as it is understood. In order to justify the 
choice of virtual mass made by Weinig, the applicable work of M. M. 
Munk, Ref . 1^, is also discussed. The formulation of the low aspect 
ratio airfoil problem and its solution by Mlliam Bollay, Ref. 16, is 
considered appropriate for discussion in Appendix I, and it is com- 
pared with the solution obtained by Weinig, 

The principal objection to the planing surface analysis by anal- 
ogy with the low aspect ratio airfoil of Weinig is that the method for 
obtaining the induced angle of attack is not clear. 

The Two-Dim ens ional Solution 

H, Wagner, whose applicable work on planing is found in Ref. 1, 
obtained a two-dimensional solution to the planing problem. As pre- 
viously stated, an inviscid, weightless fluid is assumed to exist. 
Coordinate axes are fixed in the planing surface which has infinite 
span and infinite forward length. Fluid is allowed to flow toward 
the plate parallel to the x axis shown in Fig. 2. The mass of fluid 



6. 



of thickness % contained between the free surface and the stagnation 
streamline as shoivn on Fig. 2 is deflected forward along the plate. 
The fluid below the stagnation streamline continues aft and finally 
becomes part of the wake. With the aid of the mathanatics of complex 
variables using conformal transformations, Wagner finds the normal 
force per unit width of the infinite span to be: 



^ 1 + cosr- (1 - cost)Jln ^=^~,i-Trsiiir 

2 cos'Cr 

The projection of this hydrodynamic force perpendicular to the direc- 
tion of flow gives the lift force per unit span. Then: 

L = ^Ov^ 27r sin t:" cost::' 

^ 1 + cosr- (1 - cosrUn 1-cosg ^^^^ 

2 cosr 



or 



L _ 2 7r'sin rcos r 



^^*'P^ " 1 H- cosr- (1 - cosT;)ien lz22||^rsinr 
For the infinite span case, 'r =^d and o^,. = 0. 

Application of the Momentum Theorem - Method I 

Let us now take the infinite span solution of Wagner and modify 
it for the effect of finite span. 

To begin we must put a control surface around the finite span 
planing surface so that the momentum theorem can be applied. For the 
control surface let us choose a solid shape of infinite spanwise 
width. One face is infinitely far ahead of the planing surface; one 
infinitely far aft; one at infinite depth below; and the remaining 
face at the free surface, a^bjCjdjejf of Fig. 2. 



(1) 



7. 



There is no momentum flux through the side and lower faces of 
the control surface, so we can focus our attention of the fore and aft 
faces and that part of the free surface face which cuts through the 
forward spray. 

Infinitely far ahead of the planing body, the flow is uniform 
and parallel to the undisturbed free surface, while infinitely far 
aft, a constant downwash, v,.^, is achieved. The downwash concept in 
airfoil theory is well established and is permissible in an ideal fluid 
without gravity. In the real fluid, frictional forces as well as grav- 
ity wave fonnation invalidate the discussion of a downwash infinitely 
far behind; however, what happens physically behind the airfoil or 
planing surface is not important as long as the mathematical formula- 
tion of the phenomena occurring at the body is mechanically compatible 
with the assumptions made about the fluid. 

With the established concept of a downwash infinitely far aft of 
the planing body, and no downwash infinitely far ahead, it can be 
seen that there will be a change of momentum in the negative Z direc- 
tion. Momentum flux resulting from the forward spray occurs at the 
control surface cutting through the spray, c d on Fig. 2. The time rate 
of total momentum change gives a force in the Z direction, and this 
force on the fluid is applied by the planing body. 

Let us now consider the rectangular flat surface of finite beam 
planing at velocity V. If we fix axes in the planing body, the flow 
picture is that shown in Fig, 2. The flow about the plate can be re- 
solved into two components, viz,, a component along the plate and one 



8. 



transverse to the plate. It is the transverse component of the flow 
which for the moment becomes important. The transverse component of 
flow gives rise to a virtual mass described by iifegner to be the mass 
of a semi-circular cylinder of fluid. The virtual mass is not a 
physically existing mass of fluid but it does take on physical signi- 
ficance. Appendix I describes how the virtual mass concept comes in- 
to existence J but for the present 5 suffice it to say that Wagner has 
found "Dhe virtual mass associated with the transverse component of 
flow about the planing body to be one-half that associated with the 
lifting line. M. M, Munk, Ref . 15 j shews how one finds the virtual 
mass of the lifting line by association with the potential flow solu- 
tion of transverse flow about a line immersed in the fluid. Munk's 
work is further discussed in Appendix I, Extension of the transverse 
flow about a line to the case of the planing plate is almost trivial, 
and the results lead to the mass of a semi-circular cylinder of 
diameter equal to the span. 

The force of the plate on the fluid results from the time rate 
of change of the product of the virtual mass and an appropriate velo- 
city. Similar to Weinig's low aspect ratio airfoil theory, the virtual 
mass associated with large values of wetted length-beam ratio planing 
surfaces becomes the mass of the serai-circular cylinder plus the mass 
of fluid found by j^sinr where Ji is wetted length and t the angle of 
trim (see Fig. l). 

Justification for using the virtual mass similar to that used by 
Weinig, and justification for describing the virtual mass as Weinig 



9. 



has done, may be found in Appendix I. 

From the above analysis, we find the time rate of change of 

momentum from the inlet face to outlet face of the control surface 

to be: 

F„ = m v.^^- vertical component of spray momentum (2) 

where _ 

m =(^ +Ab sinr)?V 

One should note that the F„ in equation (2) is the force of the plan- 
ing body directed vertically down (in the negative Z direction). To 
find the actual lift on the planing body, F„, it is necessary to 
eliminate the spray force. Let us define m as the mass rate of 



flow in the spray. Then 



m = ??bS 
s 



where 5 is the spray thickness (see Fig. 2), 

The lift equation by application of the momentum theorem then 
becomes: 

L = (^ + Jib sinr) €Vv^^- €VTdS V sim? (3) 

If equation (3) is divided by 2V we have 

O xr 

L ^TT /7i^ nv- . 't i*" ^VbSsinr' n\ 

2v =^^ ^T" * ^^ ^^'^^ 2V 2V ^^^ 

Transposing the spray term of equation (ii) to the left side, then 



dividing by the coefficient of v. /2V we have: 

2 V 
L - gy b S sin V i^ 

2 ~ 2V 

267^ (22^ + j^b sirn^ 



(^) 



Defining the length beam ratio as: 



E = '^ 



and the lift coefficient, C,. as: 



c, = ^ 



equation (5) becomes: 



' |?v2bA 



Z * Zl/x, sinr V 



It now becomes necessary to discuss v^^2V. At a great distance ahead 
of the planing body, we found the flow parallel to the free surface or, 
in other words, the downwash velocity zero (v. =0). At the center 
of lift of the planing body, we assume the downwash to be some frac- 
tional part of the final downwash velocity infinitely far aft of the 
bocfy. For the moment, let us call the downwash velocity at the bocfy 

k V. , where k is a fraction. From the momentum equation (neglecting 
loo 

spray), the force on the bocfy is: 

Fe 
= -m V. _, 

and the work per unit time done by the bocfy on the fluid 

Work = F k V. 

The kinetic energy per unit time associated with the downwash in- 



finitely far aft is: 



T>"ri m 2 
KE = -x V. 

2 i<» 



while at infinite distance ahead it is zero. The change in kinetic 

energy is equivalent to the work done so that: 

ATv-T-, m 2 o , 2 
Kb = 7yv. =mkv. 



From this analysis, it is evident that k must equal l/2 so that at 
the center of lift of the planing body the downwash is one-half its 
value at an infinite distance afto Now it should be clear that v,. /2V 
is the slope of the flow at the center of lift of the planing surface, 
and the angle whose tangent is v,. ^2V is called the induced angle of 
attack or induced angle of trim. Symbolically: 

V. 

tan U ^ = pTp 

Equation (6) now becomes: 

C^ + 2 f sirnr 

-^ ^ = tanO«^. (7) 

(^ + h sinr) " 

An inspection of Fig. 1 shows the relationships between the induced 
angle, 0C_. , the geometric trim angle, 7^, and the ambient flow angle, 6^ . 

r = 0^ +<j6. (8) 

e 1 

Let us now return attention to equation (1) which is the two- 
dimensional solution of the planing flat plate as developed by 
H, Wagner. The angle of trim "C" in the two-dimensional case is 
equivalent to the ambient flow angle., oC^ since the induced angle 
is zero when gravity is not considered. Replacing 'V by oi in equa- 
tion (1) and assuming oi, sufficiently small^ the following approxima- 



tions can be made: 



cos 0^ « 1 



1-cosft^ 

is fractional and its natural logarithm is negative. 

2 COS06 

e 

1-coset 

-(l-coso^ )il.n 7J y~ is positive but very small. 

e c cosofr 
e 



12. 



The denominator of (1) then becomes* 

1 + coso^^ 2 
and the lift coefficient becomes: 

2 +ysinot 

e 

Now it becomes necessary to perform some algebraic manipulation 

upon equation (9). 

C 2 7rsinoi 

L e 



2 " h * 25}^sin o^ 
e 

C, - 2 2 'yaa:fi oZ. -li-2r9ino* 
L _ ^ e e 

2 h + 2jfs±n oL 

C. - 2 - -^ 



2 + jrsin o^g 



TTsin 0^3 = cTTT^ - 2 



-Ck+ 2 Cj^ -X) 

s^ <^e = ~rTcp2i 

^^ <^e = 7^(2 - C^) 
0^ = arc sin 



e - '^'^ """ 7r'(2 - G^) 

Expanding the arc sin in series: 

2C^ 8Cj^^ 

For values of C^ ^ «85j the cubic term is less than 3% of -the first 
term and subsequent teims are much smaller. Neglecting the cubic and 
all subsequent teims of the series we have: 



13. 



From Ref . 1 an expression for the spray thickness is given as 
follows : 

s X 



I 



1 ■♦• cosy * 1 - cosT^ ^ -ysin y 
1 - cosi?"'^'^ 2 cost/ 1 - cosi 



t! *"" 2 COST/ 1 - cost/ 
Referring to Fig. 2, one notes that r is measured from the trailing 
edge to the stagnation point while J, is measured from the trailing 
edge to a line tangent to the spray root. The difference between r 
and J? is considered insignificant. If in equation (?) we now let 
tanoi:; » U. , equation (8 ) becomes 



2 G, 



r = 





2 9rsinr 


c 1 + cosr_ 

^ 1 - cos -C 


^^ 1 - costr . rsinr 
^^ 2 COS 7^ 1 - cosT^ 



n{2 - o^) 



The following substitutions are made to simplify the algebra: 

2Vsin'C 



Let 



1 + cos 

1 - COS 



r .^ 1 - cos7> ^ T/sinTf 
-€ "* 2 cos tr 1 - cos' 



= A 



(11) 



(12) 



and ^^ * ^ ^^"''^ " ^ 

Equation (12) then takes the fonn: 



tr = 



2C. 



lf{2 - C, ) * 



B 



Clearing fractions and collecting like powers of C^ we find: 

c^ - (Br+ ie + 2 - k)Q^ _ 2(A - B-r) = 

and solution by the quadratic formula yields: 



^ 



B('fc'+|) + 2 - A 



B f+ ^ B + 2 - A 



+ 2(A - BT^ 



(33) 



(lli) 



To facilitate numerical calculations, one notes that A can be simpli- 
fied considerably, Ref. 1 contains a numerically evaluated plot of 



%/r vso TTo Fran zero to about h^ degrees, the curvature is slight so 
that a straight line appiDximation will not introduce material error 
in subsequent computations » The linearization is carried out by making 
the straight line coincide with the curve at zero and 2^ degrees. The 
slope K of the straight line is then found per radian to be approximately 
0.2 so that: 

A = 2 J- sin r* 
becomes approximately 

A = (2) (0<,2)^2 = Oolfl? (15) 

Values of A and B are tabulated in Table I, then the quadratic equation 
is solved o The values found from solution of equation (lii) without 
linearizing S/j^ and those found when equation (1^) was substituted 
for A show no appreciable difference; therefore, the values of A in 
Table I are those found using equation (l5)o Table II contains the 
numerical values of equation (lit) 3 and Fig,, 3 is a plot of these values. 

Modification of the two-dimensional flat-plate-airfoil solution 
for the low aspect ratio case by the method used here led to results 
which, compared with experimental data^ overestimated the lift. Simi- 
larlyj the planing results developed here overestimate experimental 
results. 

Application of the Momentum Theorem - Method II 

The fact that planing lift results are overestimated by t^he above 
development leads one rather logically to the conclusion that the vir- 
tual mass chosen was too large Reflection upon the phenomena occur- 
ring at the plate then leads one to conclude that a new definition of 



1^. 



virtual mass based upon the ambient flow angle rather than the geo- 
metric angle of trim should be used. 

Redefining the virtual mass 5 we obtain 

L = ^VbA(^ + sin e<'^)v^^- ^V^b gsin OL^ (I6) 

Dividing by 2V, transposing, and simplifying as before, there results: 



^i-" . . C^^ . 2 I sin 0^^ 

•^p = tan U. = — — 

(^ + li sino|) 



(17) 



Substituting for sin 0^^ the value found from the two-dimensional 
solution, we find: 



C. + 2 



% 



2C, 



0^. = 



^ I 1-^(2 - C^) 



\ 



l^^ 



y(2 - C^) 
Again substituting in equation (8) there is obtained; 



2C, 



r = 



2C, 



+ - 






2^-^Cl ^ ^ e^L 



(18) 



2\ 2 r- yc 



L 



Simplification of equation (I8) using the linearization for l/jl as 
previously described leads to the cubic equation: 



,3. 



[4 



nr hY^rr 1 ^ 16 , 



If 2\ Tf 



r' 



4(¥^-^^|)^4^^ 



c,-2_%o 



(19) 



Substitution of X = in equation (I9) recovers the two-dimensional 
results. 



16. 



Solutions for equation (19) are tabulated in Table III and 
plotted on fig. Iio 

The values of C, obtained by this method underpredict the ex- 
perimental values above values of X of about 1.^ to 2, One may con- 
clude then that the virtual mass selected in this manner is too small 
for the low aspect ratio case and too high for the large aspect ratio 
case. The low aspect ratio airfoil treatment by this method also 
underestimates the lift. 

EXPERIMENTAL INVESTIGATION 

Purpose 

A comparison of experimental data with the results computed by 
the two formulas for the lift coefficient developed herein was under^ 
taken. It soon became evident that existing published data did not 
cover the entire aspect ratio-angle of trim range desired; therefore, 
the necessary tests were performed in the towing tank to supply the 
otherwise unavilable data. To this end, the facilities of the sea- 
plane tank at the Experimental Towing Tanky Stevens Institute of 
Technology were employed. 

Equipment and Procedure 

The model used for the tests was machined from half-hard brass 
bar stocks and provision for supporting the model in the towing car- 
riage was made by mechanical linkages shown in Fi^. 5" Th© overall 
length of the model was 18 inches, the beam 2 inches ^ giving the 



17. 



general appearance of a rectangular ski. The lengthwise edges of the 
upper surface of the ski were beveled at an angle of 30 degrees as 
shown in the sketch of Fig, 5. The upper surface bevel provided 
sharp edges for a clean break-away of flow. The flat bottom which 
was the wetted surface during the tests was polished and marked with 
Dykem Steel Bluing to permit ease of reading the wetted lengths frcm 
the \inderwater photographs made during each test run. The marking de- 
tails are also shown in Fig. 5. To eliminate the adverse effects of 
spray on the carriage, it was found that vertical separation between 
the model and carriage supporting linkages was necessary. To provide 
the required vertical separation, two mounting rods, approximately 
3-l/i; inches long were tapped into the upper surface of the model 
near the after end. 

The seaplane tank where the tests were conducted is 313 feet 
long, 12 feet wide, and 6 feet deep when the water level is just even 
with the opening in the standpipe inside the tank. A maximum carriage 
speed of 50 feet per second can be attained with the selected speed, 
amplidyne controlled, carriage driving motor. A tachometer generator, 
the armature of which is driven by the carriage driving motor shaft, 
supplies electrical infonnation to the field of the amplidyne system 
to provide speed control. With a selected voltage, representing a 
certain speed, the carriage speed can be maintained to within O.06 
feet per second. A General Electric photoelectric tachometer records 
the carriage speed. 



18, 



The camera for making underwater photographs of the wetted sur- 
face was a specially adapted Beattie Varitron, Model E, Data Record- 
ing Camera. The Varitron is a low speed, pulse type camera, the operar- 
tion of which is completely automatic. An electrical signal operates 
the shutter of the camera, and as the shutter opens, it triggers the 
photoflood lights to make the exposure. Following the exposure, the 
shutter is automatically disconnected while another circuit provides 
power for winding the film in the magazine to the next frame. When 
film winding is completed, the shutter circuit is re-engaged, and the 
apparatus is ready to make another exposure. A recording chamber on 
the side of the camera automatically records sequence numbers and in- 
formation on a platten to identify the test. The camera for the tests 
described herein was housed in a watertight box and pressurized to in- 
sure against leakage. A control for re3:Qote focusing after the camera 
had been positioned in the tank was also providedo A light source, 
mirror, photocell, and thyratron were used to actuate a relay which 
delivered the electrical pulse to the camera shutter. 

The model was installed in the overhead towing carriage as shown 
in Fig. 5 with provisions made to set the angle of trim as desired. 
With the trim angle set at zero degrees, adjustments were made to in- 
sure that there was no angle of roll or yawo By gently lowering the 
ski into the water and observing the bottom surface with a mirror held 
underwater below the ski, a simultaneous wetting of the entire bottom 
surface indicated proper alignment in rollo law was checked by plac- 
ing a bar with reference marks inscribed thereon at the front edge of 



the ski, perpendicular to the direction of motion and in the plane of 
the bottom surface of the ski. Then, if by moving the carriage for- 
ward, the edge of the ski remained on the reference mark, yaw angle 
was considered zero. A spirit level protractor was used to set the 
angle of trim. 

With the desired angle of trim set, a calibration in heave was 
made so that during the runs, a first order approximation on the 
wetted lengths could be ascertained. To accomplish the heave cali- 
bration, the ski was again gently lowered into the water and observa- 
tions in the underwater mirror were made similar to those for the roll 
adjustment. When the trailing edge of the ski just touched the water, 
the heave reading was taken on the large scale as shown in Fig. 5» 
This reading was listed on the data sheet (Table IV of this report) 
as the heave reference, and this reference value was subtracted from 
the heave reading made during the run. The difference (or sum of ab- 
solute values in the case of a negative heave reference) gave the 
vertical projection of the wetted length of ski excluding the wave 
rise. Then by an empirical wave rise factor, described in Ref . 18, 
the approximate total wetted length could be found. Let it be empha- 
sized here that the heave readings were used in the tests as a control 
measure only. Since there are wave disturbances in the tank after the 
first run, slight transient effects limit the accuracy of heave read- 
ings. The procedure for obtaining approximate wetted lengths from 
heave readings was as follows (see Fig« 6): 



Let: -g- = wetted length/beam ratio of the ski neglecting 
wave rise 
h = difference in heave (mnning heave minus reference 

heave, inches) 
^^ * empirical wave rise factor = <,3 
"C = trim angle (degrees) 
X = J^ = total wetted length/beam ratio 

b b b b sinf I' 

^ = b = rilrt * -3 h = (X - .3Xb Sim;) 

Anticipating specified wetted lengths made it is possible to 
determine approximately the desired value of h so that during the 
runs, the values of h served as a check on actual wetted lengths 
being obtained. Adjustment of the load to get the desired value of 
h and, thus, the actual wetted length, was then practical. 

In order to eliminate all effects of gravity on the lift co- 
efficient obtained by the tests, all runs were made with a speed co- 
efficient, Cy, at a nominal value of twelve. Theoretical justifica- 
tion for independence of gravitational effects on the total lift coeff- 
icient at high Froude Numbers will be treated later, (Refer 
to Fig. 7)0 Since the beam of the model was two inches, a carriage 
speed of 27»8 feet per second was found to be appropriate. (Note that 
C^ = V/Vi5 or V = Cy \/gB). 

For each run, a pre-determined load was applied to the model 
which was allowed to run free in heave with a fixed angle of trim and 



at constant speed » The wetted length for each run was photographed 
so that correlation with load could be made for data reduction. 

Table IV, which is a copy of the laboratory data sheet, contains 
all of the information pertinent to each of the test runs. Table V, 
containing wave rise "correction" factors computed from the heave 
readings and the underwater photographs, was made as a check on the 
previous empirical wave rise factor and is intended to supplement 
present existing information for future tests which other investiga- 
tors may perform in the aspect-ratio-angle of trim range covered 
herein. 

Experimental Results 

Reduction of experimental data from the photographs taken and 
carriage speeds recorded are found in Table VI, 

Since most of the experimental data collected from the tests 
here were b^ond the range of existing data, it was not possible to 
evaluate the reproducibility of such results, particularly in the 
limited time availableo The general trend of curves faired through 
the test points obtained in this investigation is to lie between the 
curves of Fig. 3 and k which leads one to believe that the results are 
reasonable. It is noted from the ratios of computed values based ono^ 
to experimental values tabulated in Table VII and shown on Fig, 8 that 
the data collapses fairly well for trim angles of 18 degrees or less. 
Above 18 the dependence of experimental values of the lift coeffi- 
cient upon the angle of trim manifests itself, but it is not clear 



22, 



from a physical viewpoint just what the relationship between \, 7^, 
and C, should be. 

GQLPARISQM OF THEORETICAL RESULTS WITH EXPERE/IENTAL DATA 



General Discussion 

In order to detennine the validity of calculations made by cor- 
recting the two-dimensional planing solution for the effect of finite 
span, a comparison was made with experimental data. The principal 
source of data used was from Weinstein and Kapryan, Ref. 19^ Since 
the existing data available did not cover the complete angle of trim, 
wetted length-beam ratio range desired, additional data were obtained 
experimentally and are presented herein. Of the several methods avail- 
able for comparing data, it was decided that ratios of computed values 
to experimental values plotted against X with fas parameter would 
give the most enlightening results. Values of the ratios using both 
method 1 and method 2 previously described for computing values of 
C. are found in Table VII and are plotted on Fig. 8 and 9« Since the 
ratios are not of the order of unity, it becomes necessary to make an 
empirical correction to the computed values in order to predict the 
lift coefficient satisfactorily. 

Empirical Correction to the Curves of C, Computed by the Cubic Equation 

For m defined as a function of 0^ , it is seen that as X becomes 

e' 

greater than about 2, computed values of G^ from equation (19) imder- 
estimate the experimental values. The ratios of computed C^ to ex- 



23. 



perimental Cj values found in Table VIII and plotted on Fig, 8 show 
that for trim angles of 18 degrees or less, the ratio plot tends to 
collapse into a single curve. For the case of C- computed on the basis 
of equation (lli) the plot of ratios, Fig, 9j does not collapse as neat- 
ly as the plot of Fig. 8, For this reason, empirical correction to the 
curves of G, computed by equation (19) was elected. 

The curve shorni on Fig. 8 was faired to fit most of the points 
and the empirical factor found as follows: 

L computed 

Q 

L experimental _ , /„ -, 

K 

where yi = empirical correction factor. 

For 2°^ 7:'^l8°, >^ = f (X) and we call this factor y^^. For 
18°^ 7:^^30°, >(= t(\,lf) and this factor is denoted by yi^. 

There are, of course, several suitable functions to fulfill the 
requirements on h, but the ones chosen here are: 

\ = 1,3^9 - tanh (^^) (21) 

7^2 ="^1 + i^ ^anh X^ (22) 

From Fig. 8, the intercept value of the curve is recovered from >^, 

when X = 0. 

o _ 18° 
In equation (22), Al^is defined as ' ^^„ — radians. Then when 

\- 0, \= \, 

^he empirical factors, Y), and >7p, have been applied to values of 

C- computed by equation (19) and the results tabulated in Table VIII, 

Fig, 10 is a plot of the empirically corrected values of Ct and is 



21,. 



the one to be used for estimating the lift coefficient when gravity- 
is not considered. 

Fig. 11 shows the empirically corrected curves of Ct vs. X with 
experimental data points plotted thereon. 

THE EFFECT OF GRAVITY. 

One can recognize that the influence of gravity in the case 
of a planing body can manifest itself in two separate ways. One 
part of the gravity contribution is a static force which is the 
Archimedian lift or buoyancy; the other is a cfynamic effect which 
attends the phenomena of the development of waves. 

Until recently, the effects of gravity on the dynamic lift have 
been practically ignored in planing lift calculations, partially be- 
cause the mechanism of the dynamic effects has not been clearly under- 
stood (Ref. S, 17, 20), and partially because the theory, including 
the dynamic effects, is quite complex and difficult to evaluate. 

In 1937, L, I. Sedov, Ref. 2, presented a two-dimensional asymp- 
totic solution to the planing problem, including the effects of 
gravity on the dynamic lift in the region of high Froude numbers. 
For more practical applications, Hajime Maruo, Ref. 3, published in 
1951 a two-dimensional solution to the planing problem, and followed 
this in 1953 J Ref. 21, with a solution for the finite span. 

It is considered appropriate to summarize here the Maruo paper, 
Ref. 21, which reveals the effects of gravity on the lift of the 
planing flat plate of finite span. 



25. 



Llaruo assumes a form for the velocity potential function ^ which 
involves an unknown pressure distribution over the planing surface. 

o 

This potential function is constructed to satisfy the free surface 

condition of constant pressure except on the planing surface. Then, 
from the potential function, a general formula for the downv/ash is 
found. _« •© ■) 

8 -rr o 

Maruo points out that the downwash can be separated into two com- 
ponents; one corresponds to a profile characteristic and is present 
even in the two-dimensional flow; the other is a downwash arising 
from the effect of finite span. 

The downwash condition applied at the planing surface to satisfy 
the condition of no flow through the plate, gives rise to an integral 
equation for the unknown pressure distribution over the plate. 

Maruo, by a procedure analogous to Glauert's solution of the 
finite span airfoil problem, simplifies and solves the integral equa- 
tion for values of X ^1/2, or for aspect ratios greater than about 
two. Figo 7 is a graphical representation of the solution when X— ^0. 

One should note on Figo 7 that there are two curves, I and II, 
Curve I is the plot of a numerical solution to the exact equation 
while II is a plot of the simplified equation evaluated analytically. 
Attention is directed to the shape of the curve of Fig, 7 which shows 



26. 



that the theoretical value of C^ , not considering buoyancy effects, 
rises from a C^ of 0.7^ and approaches asymptotically a constant value 

at higher C„. Most plots of experimental data show the total lift co- 

1 2 
efficient without subtracting the coefficient of buoyancy 3/^ ^V b / 

from the data^ therefore, the effect of gravity on the dynamic lift 
appeals to be a monotonically decreasing function of ^„ rather than 
an increasing function as shown by the theory. 

The practical results of Maruo's work are given by Equation (1^2) 
of Ref . 21 which, when transcribed to the notation used herein, be- 
comes : 



where ]^ is a function of the speed coefficient, C„, 

A graph of the T^ function, given in Ref. 21, can be linearized 
to the following approximation: 

^= 0,667 + %^. 



Then C^ becomes 



'Y 



If 

\ " 0.667>^ ^^ 0.333 + Q.233>^ ^^^^ 

G 2 



From equation (23), it is evident that the lift coefficient becomes 
independent of the cfynamic effects of gravity for 



. / 0.7X 



v^ yi -^ 2x 

One should be aware of the fact that the mathematical validity of 
equation (23) was based on \ <rl/2o 



27. 



For small X (such that 2X<-< 1), it is seen that the first tenn 
of the binomial expansion of the inequality yields C„> 0.6V^which 
agrees quite well with the empirical value of C /V^>1 established 
by Daniel Savitsky, Ref. 18, as a criteria for independence of gravity 
in the flow. 

The effect of gravity on the dynamic lift from this analysis is 
seen to become negligible at low values of Cy. Those investigators 
who insist that the value of G„ above perhaps 10 is necessary to elim- 
inate the gravity effect refer to the combined buoyant and dynamic 

effects. The dynamic effects become small at low values of C^., but 

2 
until the dynamic pressure, l/2oV , becomes large enough to make the 

ratio of static buoyant force to dynamic pressure a small percentage 

of the total lift, buoyant effects will persist. 

For most practical cases, the lift coefficient, as Cy approaches 

the order of 10, is clearly independent of all gravity effects. If 

one subtracts from the total lift coefficient the buoyant lift con- 

B 
tribution defined by T~Z „2, p, it is seen that the remainder is in- 
dependent of all effects of gravity only for C > 0.6V5vrwhen X ^ 1/2. 



SUmyLARY AND CONCLUSIONS 



The results of this investigation show that theoretical evalua- 
tion of the effect of a finite span on the planing surface problem is 
quite complex. To remove some of the complexity, the problem was con- 
sidered in two phases. First, the effects of gravity were neglected, 
and an ideal fluid assumed; then second, the effects of gravity were 
discussed. 



28. 



Neglecting gravity, equation (ll|) was obtained from a combination 
of the two-dimensional solution and application of the momentum 
theorem to a selected virtual mass. Equation (19) was found in a 
similar manner but with a different virtual mass. Both equations (lli) 
and (19) overestimate the lift coefficient for aspect ratios greater 
than about one-half which indicates that the choice of virtual mass 
for that range of aspect ratios is too high. Equation (I9) underesti- 
mates the lift when wetted length-beam ratios are greater than about 
two (aspect ratio less than one-half), while equation (lii) overesti- 
mates the lift for all length-beam ratios. 

Fig. 1 shows cross-sectional areas of the "volumes" containing 
the virtual mass for each case. Tables II and III give tabulated 
values of the calculations made by each of the above methods, and 
Table VIII gives the empirical corrections to make values computed 
by equation (I9) agree with experimental data. JEkpirically corrected 
values of the lift coefficient are plotted on Fig. 10, and comparison 
of these curves with experimental data are shown on Fig. 11. 

While the analytical results obtained are considered reasonable, 
empirical correction to the theoretical analysis was required to get 
better agreement with experimental data. Thus, the development can be 
considered as a semi-empirical method for evaluating the effect of a 
finite span on the solution to the planing problem in a weightless 
fluid. 

For the analysis made independent of gravity at length-beam- 
ratios greater than about 2, the method leading to equation (19) 



29, 



indicates that the virtual mass may be too small. The virtual mass 
associated with equation (lii) may be too large. The most prominent 
physical explanation for the inconsistency between theoretical and 
experimental results is that the downwash along the wetted length of 
the body is not of constant magnitude. Bollay, Ref. 16, indicates 
that free vortex lines are shed along the chord of the low aspect 
ratio airfoil so that at the trailing edge^the contribution to down- 
wash tnere is not only from the vortex wake, but also from the shed 
vorticity ahead. This, in effect, produces a curvature of the flow 
from leading to trailing edge which has the effect of introducing an 
apparent camber to the foil. 

For the low aspect ratio (large \) planing surface, the curvature 
effect may have a significant influence on the lift, and is offered as 
a possible explanation for the difference between predicted and experi- 
mental values . 

It is worthy of notice here that the technique similar to that 
used for modifying the infinite span airfoil for the effect of finite 
span is a logical procedure to use for the planing surface. As 
pointed out by many other investigators, the flow for most practical 
purposes can be considered ideal and weighless in the planing regime, 
and from the stagnation point to trailing edge on the lower surface, 
both airfoil and planing body have almost identical pressure distrib- 
utions. 

The consideration of gravity led to results which show that the 
effect of gravity on dynamic lift is a function of the speed coefficient 



30. 



and the wetted lenpth-beam ratio. It was found that for all practical 
purposes, the effect of gravity on dynamic lift could be disregarded 
for G ;p^ 0.6V5rwhen \'<Zl/2. Buoyant effects remain significant 
until the dynamic pressure reaches such a value as to make the co- 
efficient of buoyancy very small compared with the dynamic lift 
coefficient. 



31. 



BIBLIOGRAPHY 

1. Korvln-Kroukovskyj B. V. : Hydrodynamic Design of Seaplanes II, 
Experimental Towing Tank, Stevens Institute of Technology, 
Hoboken, N.J. (Summary of the applicable work of H. liVagner 
included ) . 

2. Sedov, Lolo- Two-Dimensional Problem of Planing on the Sur- 
face of a Heavy Fluid, Unpublished Translation by B.V. Korvin- 
Kroukovsky, Experimental Towing Tank, Stevens Institute of 
Technology, Hoboken, N.J., 1937. 

3. Maruo, Hajime: Two-Dimensional Theory of the Hydroplane, Pro- 
ceedings of the First Japan National Conference for Applied 
Mechanics, 195l» 

ii. Korvin-Kroukovsky, B.V. : Lift of Planing Surfaces, Journal of 
Aeronautical Science, (Readers' Forum), Vol. 17, No. 9j September 
1950, pp. 597-599. 

5. Perry, Byrne: The Effect of Aspect Ratio on the Lift of Flat 
Planing Surfaces, Report No. E-2ii<,5 (Contract N6onr-2lili2li, 
Project NR 23J4-OOI), Hydrocfynamics Lab., California institute 
of Technoloi^r, September 1952 o 

6. Perry, Byrne: The Lift of Flat Planing Surfaces, Preliminary 
Report No. N-3I.35 Hydrodynamics Laboratory, California institute 
of Technology, March 1950. 

7. Bollay, William: A Contribution to the Theory of Planing Sur- 
faces, Proceedings Fifth Int. Cong. Appl. Mech. (Cambridge, Mass., 
1938), John Wiley and Sons, Inc, ^ 1939^ pp. hlh-hn ' 

8. Perring, W.G.A. and Johnston, L. ; The Hydrodynamic Forces and 
Moments on Simple Planing Surfaces, and an Analysis of the ifydro- 
dynamic Forces and Moments on a Flying-Boat Hull, R & M No. I6J46, 
British Aeronautical Research Committee, 1935* 

9. Sottorf , W. : Analysis of Experimental Investigations of the 
Planinp Process on the Surface of iVater, NACA T.M. No. I06I, I9I1I1. 

10. Landweber, L. and Eisenberg, P.: Characteristic Guides for 
Planing Surfaces, David Taylor Model Basin Report R-80, 191^3 . 

11. Locke, FoW.S., Jr.: An Bnpirical Study of Low Aspect Ratio 
Lifting Surfaces with Particular Regard to Planing Craft, 
NAVASR DR Report No. 10ii3, January 19li8. 



32, 



12, Shuford, C,L. , Jr.: A Review of Planing Theory and Experiment 
with a Theoretical Study of Pure Planing Lift of Rectangular 
Flat Plates, NACA T„N. 3233, Langlqy Aeronautical Laboratory, 
August 195^0 

13. Siler, William: Lift and Moment of Flat Rectangular Low Aspect 
Ratio Lifting Surfaces, Experimental Towing Tank Technical 
Memorandum No, ^(:>^ Unpublished, 19ii9» 

Ih. Weinig, F. : Lift and Drag of Wings with Small Span, NACA T.M. 
No. 1151, August 19ii7. 

15 c Munk, Max M, : Fundamentals of Fluid Dynamics for Aircraft 
Designers, The Ronald Press Co,, New lork, 1929. 

16. Bollay, William: A Theory for Rectangular Wings of Small Aspect 
Ratios, Journal of Aeronautical Sciences, Vol. IV, No. 7, May 
1937. 

17. Weinig, F. : On the Theory of Hydrofoils and Planing Surfaces, 
NACA T.Ivi. No, 81i^, January I938. 

18. Savitsky, Daniel and Neidinger, Joseph W, : Wetted Area and 
Center of Pressure of Planing Surfaces at Very Low Speed Co- 
efficients, Experimental Towing Tank, Stevens Institute of 
Technology, a Sherman M, Fairchild Publication Fund Paper by 

the Institute of Aeronautical Sciences, Deptember 195^j No. FF-11. 

19. Weinstein, Irving and Kapryan, Walter J, : The High Speed 
Planing Characteristics of a Rectangular Flat Plate Over a 
Wide Range of Trim and Wetted Length, NACA T,N. 2981, Langley 
Aeronautical Laboratory, July 1953 • 

20. Wagner, H. : Planing of Water Craft, NACA T,Mo No, II39, 
April 191480 

21. MaruOj, Ha jime : Theory of the Hydroplane with Broad and Short 
Planing Surfaces, Bulletin of the Faculty of Engineering, 
Yokohama National University, Yokohama, Japan, Marcy 1953» 

22. Milne- Thorns on, L.M, : Theoretical Aerodynamics, MacMillian Co., 
Inc., New York, May 191^7. 

23. Perelmuter, A. ^ On the Determination of the Take-Off Character- 
istics of a Seaplane, NACA T.M. No. 863, May 1938. 

2[i, Winter, H. : Flow Phenomena on Plates and Airfoils of Short Span, 
NACA T.M, 798, 1936, 



33. 



2$. Lawrence, H.R. : The Lift Distribution on Low Aspect Ratio 
Wings at Subsonic Speeds, Journal of Aeronautical Sciences, 
Vol. 18, No. 10, October 19^1. 



3ii. 



APPENDIX I 

General 

The use of incompressible potential flow theory to treat the 
effects of low aspect ratio on wing lift has been carried out for 
the case of a flat plate airfoil in at least two distinctily different 
ways, but with essentially the same result. The two different methods 
cited are: 

(1) The solution by F. Weinig, Ref . lli, whereby the momentum theorem 
is applied with an estimate of virtual mass, and an estimate for 
the induced angle of attack, ^^, is made based on cascade theory, 

(2) The integral equation solution of W. Bollay, Ref. 36, in which 
the boundary condition — that there be no flow through the 
plate — is satisfied by equating the transverse velocity com- 
ponent of the stream, V sino", to an integral expression estab- 
lished for the downwash at the wing. 

It should be noted that both of these methods are developed for 
an airfoil in an incompressible fluid neglecting gravity; therefore, 
it is reasonable to expect the results to be analogous to that por- 
tion of the planing problem which is treated independent of gravity. 
Wagner, Perlemuter, Perry, and Shuford, Ref. 1, ^5 6, 12, 20, and 23, 
have indicated that the integral of pressures over the wetted surface 
of the planing body, excluding the region about the leading edge 
forward of the stagnation point, is almost identical with the pres- 
sure distribution over the same portion of a flat plate airfoil; 



35. 



thus, the planing surface-airfoil analogy seems to be reasonably- 
founded. 

It is not the purpose of this discussion to present the details 
of the airfoil solutions of Weinig and Bollay, but rather to show 
how they lead to a better understanding of the virtual mass chosen 
to modify the two-dimensional planing surface solution for the effect 
of a finite span. The fact that the solutions of both Weinig and 
Bollay lead to essentially the same result is graphically revealed 
in Fig. 12. The experimental points shown in Fig. 12 were obtained 
by H. Winter, Ref. 2I4, and are included to show the relative merit 
of each of the theoretical results displayed. One obvious reason for 
the difference between the theoretical solutions of Weinig and Bollay 
is that mathematical complexity forced Bollay to make several simpli- 
fying assumptions in order to solve the integral equation established 
in his fonnulation of the problem. Although Bollay' s treatment of 
this problem seems to be established on a more rational foundation, 
Weinig' s results correlate better with experimental data as pointed 
out by HoR. Lawrence, Ref. 2^. 

Virtual Mass 



Modification of the two-dimensional planing surface problem for 
the effect of a finite span by application of the momentum theorem 
makes necessary the choice of an appropriate virtual mass. Attention 
is directed to the low aspect ratio airfoil solution by YiTeinig wherein 
the virtual mass selected is composed of the virtual mass associated 



36. 



with a lifting line plus the streamwise projection of the chord 
bJcsin oC„ In order to better \mderstand Weinig's choice of virtual 
mass, a review of the salient features of the virtual mass concept 
as given by MoM. Munk, Ref. 1^, are considered. A brief review of 
the most important steps leading to the definition of virtual mass 
follows . 

If an incompressible fluid at rest is given an impulse which re- 
sults from the action of normal pressures alone (frictional forces 
assumed zero) the resulting flow is potential flow provided no dis- 
continuities exist and all spacewise partial derivatives of the poten- 
tial function are continuous » If it is assumed that the impulsive 
force creating the flow acted for some finite distance, work would be 
done on the fluid, and since the fluid was originally at rest with 
zero kinetic energy, the work done represents the kinetic energy 
stored in the flowo It is convenient to replace the total energy 
of the fluid particles moving in many different directions with the 
equivalent energy of a certain geometrically defined mass of the fluid 
moving with the velocity of the bodyo This assumed mass, introduced 
for mathematical convenience, is labeled virtual mass and may be 
different in magnitude for different directions,, depending on the 
velocity chosen in its definition. 

If a very thin airfoil with axes fixed in the body moves with 
a steady velocity in air which was originally at rest, velocity about 
the airfoil at angle of attack,©^, can be resolved into two components, 
one nomal and one parallel to the foil as shown in the sketch. 



37. 




In an ideal fluid (no friction), 
the component parallel to the air- 
foil does not contribute to the 
forces. Attention is then concen- 
trated on the normal component which 
is referred to as the transverse 
flow component,, The line pq on the sketch represents an element of 
the airfoil of span bo (It has been shown that when the span to chord 
ratio is large, the entire foil may be represented by such a line,) 
The wake at some distance behind this airfoil behaves like a flat plate 
of span b moving with velocity v perpendicular to the plane described 
by the plate. The complex potential for the flew net about this hypo- 
thetical flat plate is givai by: 

w - -^ iyr - iv (z - \lz^ - (|}^) (Al-1) 

where z = x ■*■ iyj v = transverse velocity component^ and b/2 = semi- 
span. The product, p0» can be thought of as the impulsive pressure 
necessary to create the flow field in which kinetic energy is storedo 
Then, the expression for kinetic energy in terms of the potential func- 
tion is : 



sj oefS^ds 



where O = fluid density 

d0/dn = component of transverse flow velocity 
ds ~ surface element over which the integral is performedc 
This is the total kinetic energy of the flow, and as previously 



(Al-2) 



38. 



stated, it can be expressed in terms of the virtual mass, m, as 

T = 1/2 mv^ (Al-3) 

where v = d0/dn = constant on the surface of the plate. The total 
kinetic energy of the flow as foiind by the potential function is 
equated to the energy expressed by the virtual mass so that 

Since o = constant and d{^/dnj = v = constant, there is obtained: 

Jplate 



vol = ij ^S (Al-5) 

This defines the volume of fluid of density p associated with the 
transverse velocity. It should be noted that the virtual mass is 
not a physical body of particles; its properties ^ as stated previously, 
are directional such that for a direction other than that of the trans- 
verse flow, the virtual mass may be entirely different. 

On the surface of the h^rpothetical plate (i.e. y = in the com- 
plex variable notation): 



■•vP- 



= V Yl - x'=^ (Al-6) 

Substituting the value of from equation (Al-6) into equation (Al-5), 
and noting that djMn is constant over the plate equal to v, the 
volume of fluid defining the virtual mass is obtained: 

Vol = y I - x^ dx (Al-7) 

Since the surface over which the integral is performed extends 
from -b/2 to +b/2 on both upper and lower surfaces of the plate, the 

final expression for the volume becomes 

/b/2 r~2 ' 2 

r/| - x^ dx =-^ (Al-8) 

-b/2 



39. 



This is the volume of fluid containing the virtual mass and is shown 
here to correspond to the mass of a circular cylinder of diameter 
equal to the span of the plate. 

The Lifting Line 

In the previous derivation of lift over the span b and an element 
of length dJ?, the downwash velocity, v = d0^/dn was necessarily con- 
stant along the span and was assumed to have no chordwise variation. 
The same constant downwash velocity is known to exist in the case of 
the vortex system consisting of bound and trailing vortices such that 
the circulation of the bound vortex varies elliptically along the 
span. The bound vortex is known as the Lifting Line, and it can be 
taken to represent the entire wing when the ratio of span to chord 
is large. 

For multiple lifting lines disposed vertically in close proximity 
to each other, Munk has indicated that the interference existing be- 
tween the flow patterns associated with each line reduces the virtual 
mass of the single lineo The mathematical complexity involved for 
the case of three or more lines limits the discussion here to only 
qualitative, order of magnitude results. 

In the case of a biplane, two lifting lines separated by a small 
vertical distance, h, can be assumed to replace the wings. The re- 
sulting volume containing the virtual mass associated with the trans- 
verse flow is given by Munk as approximately the cylinder of diameter 
equal to the span plus the rectangular mass enclosed between the upper 



and lower wings: _ 

Vol =2^ + bh 

The Effect of a Long Chord 

The virtual mass associated with the lifting line, used in 
application of the momentum theorem to find the lift for a finite 
span airfoil leads to the prediction of reasonable values for span 
to chord ratios which are large. As the chord lengthens with respect 
to the span, the single lifting line representation becomes invalid, 
principally due to the fact that the free vortex lines no longer 
leave the trailing edge in a flat plane of the main stream, but in- 
stead are shed along the chord forward of the trailing edge to form 
a three-dimensional vortex array in the wake. 

In order to satisfy the boiindaiy conditions d0/dn = V sino^ 
on the surface of the airfoil, and of the streamlines flowing smooth- 
ly off the trailing ed^e, Bollay, Ref. 16,22, has assumed a distribu- 
tion of bound vortex lines along the chords After certain simplifi- 
cations, Bollay has solved the integral equation for the distribution 
of circulation among these vortex lines (or lifting lines). 

Interpreting Bollay 's results in terms of virtual mass, the 
chordwise spacing of ar^r pair of bound vortex lines at a distance d* 
apart leads to an equivalent vertical spacing of d£sino(.. Consider- 
ing this in the light of Munk's theory of the biplane and multiple 
lifting lines, the integral along the chord of the contributions of 
all vortex pairs gives the contribution bJcsino6to the volume of the 
virtual mass as an approximation. The total volume associated with 



ia.\ 



an airfoil of small span and long chord then becomes "t — + bfcsinoi. 
It is observed that this is the exact expression chosen by Weinig. 
It should be noted, however, that od in this case needs an inter- 
pretation,, Weinig has assumed it to be equal to the geometric angle 
of attack of the airfoil, but in the spirit of the above derivation, 
it would appear more logical to use the ambient flow angle in the 
immediate vicinity of the airfoil rather than the geometric angle of 
flow. 

Modification of the two-dimensional airfoil solution, using the 
momentum theorem, with the virtual mass based on geometric angle of 
attack, then on the ambient flow angle, follows. 

Modification of the Two-Dimens ional Airfoil Lift Coefficient for the 
Effect of FinJte Span 

Method 1 - virtual mass based on geometric angle of attack,©^. 
From the momentum theorem: 

oyfot 



L = A V. = &^ (f^ X sino<)v. (Al-9) 



where m = (rj~ + b/^sinodj^:^ 

b 
From energy relationships: 

V. 

7~ = tan£^^(V„ (Al-10) 

Dividing (Al-9) by 2V, substituting the result of (Al-lO) and solving 



forO^^, , there is obtained 

" (2V)(eVb.£)(f + X sino^/ TT^ UK sino,^ 



QC, = ^^i— — - = ---_il^_-^ (Al-11) 



a2. 



where C, = 



^ le^^u 



Now: 



Q a. 

and from two-dimensional theory: 

\ = 2Trsino^^ co^^ 
Dividing (AI-I3) byTT: 

— r = 2 sin(y coso^ = sin2o^ 
Jf e e e 



2^ = arc sin — 
e 77- 

Vi^ritinp the series expansion for arc sin 



TT- 



*o = l^ 



I • &r i • 



Neglecting all terms in the series higher than the cubic, and sub- 
stituting from (Al-16) and (Al-11) into (Al-12): 



2 I 77" [jij 6 ( 7r+ ^^ sinQ6 



Solving : 



.3 1^ , 125^ 



C^ - 12WoC= 



'L '!"" ■ 77+ hX sino6| L 

Values of Q^ substituted into (AI-I8) show no appreciable dif- 
ference from the results obtained as follov^s. 
In equation (AI-I3) let: 



Then (AI-I3) becomes: 



cos oi :::i 1 
e 

sinO^ ^^ 
e e 



L e 



(Al-12) 
(Al-13) 

(Al-lit) 
(Al-15) 



(Al-16) 



(Al-17) 



(Al-18) 



(Al-19) 



ii3. 



Multiplying (Al-12) by 27/7 then substituting from (Al-11) and (Al-19), 
there is obtained: 

2 7rxc 

2iroi = C + ■ ^ |. ■. — , (Al-20) 

Equation (Al-20) simplifies to: 

TT + ijX sinoi 
Values of C_ computed by (Al-21) are tabulated in Table IX and plotted 
on Fig. 13. It is observed that this method overestimates the ex- 
perimental values of Winter, 

Method 2 - virtual mass based on ambient flow angle, o^ . 
Similar to method 1, there is obtained forO^. : 

xc ■^• 

Oi. = ^^ 1..^ ..^^ (Al-22) 



jr ■¥ \\K sino^ 



Approximating again: 



cos 0^ ~ 1 
e 

sinO^ fi.0^ 

\ e e 

and substituting©^ = 'orr^ ^^® equation: 

e 1 
there is obtained: 

Simplifying and collecting coefficients of like powers of G : 



■^.■r 



=l' 'm * '"'' - ^'^K - '^ = (A1-2W 



hh. 



Equation (Al-2[i) is solved by the quadratic formula, values tabulated 
in Table IX and plotted on Fig. I3 for comparison with method 1 and 
the experimental data of Winter. It is observed that C^ is underesti- 
mated by equation (Al-2l^). 



APPMDK H 
Foundations of the Theory of Planing 

Translation of Paragraph 75 of the Russian Textbook, Water Resistance 
to Movement of Ships, by P„ A. Apuchtin and J, I. Voitkunsky, 
Moscow, 19$3» 

The viscosity of a fluid affects the pressure distribution along 
the planing surface very little and separation of the boundary layer 
is not observed on planing surfaces. This provides the basis for in- 
vestigation of the phenomena of planing by methods of hydrodynamics of 
a perfect fluid, in particular the potential flows. 

In the domain of the theoretical investigation of planing, the 
Soviet scientists played a leading part. In 1929, G. E. Pavlenko 
first obtained the solution for the lifting force and resistance of 
a plate of infinite span, planing at a small angle of trim on the sur- 
face of a perfect fluid. 

Later, G, E. Pavlenko developed the foundations of the theory of 
planing, taking into account the finite span^ studied the physical 
phenomena connected with planing, and also investigated the question 
of recalculation of model test results of planing surfaces to full 
size. 

In subsequent works, the two-dimensional problem of planing of 
plates on the surface of a perfect fluid was attacked along two paths: 

1) by the methods of streamline flows, not taking into account 
the weight of the fluid. 



hs. 



2) by methods of the theory of waves of small height taking into 
account the weight of the fluid. 

The investigations of planing by means of streamline flows for 
infinite depth and for finite depth of a fluid were conducted by the 
academicians S. A, Chaplygin, Mo I. Gurievich, A. P. Yampolsky and 
Wagner. 

The solutions of the planing problem of a curved plate, taking 
into account the weight of the fluid were given by the academicians 
N, E. Kotchin and L. I. Sedov. 

The planing of the surfaces moving one after another (in tandem), 
interesting in connection with the action of steps, for the case of 
gravity-less fluid was investigated by L. I. Sedov. 

Let us consider the origin and the calculation of water resist- 
ance for the planing flat plate on the surface of a perfect fluid. 

Large pressure gradients are observed along the lower surface in 
the movement of the plate. Plots of the pressure distribution, 
measured experimentally along the centerline of the plate and along 
its edges at various angles of trim, are shown on Figo I3J4. Along 
the span of the plate pressure changes are not significant, while 
normal to the span in the region of the leading edge the pressure 
gradient is very large<, A large drop of pressure is observed at the 
lateral edges, where the pressure abruptly drops to a value corres- 
ponding to the pressure in the undisturbed fluid. 

in regions of large pressure gradient (in the regions of inter- 
section with the free water surface), spray jets are formed. The pro- 

(1) See Figo lii 



hi. 



jection of the reaction of the jet, in the direction opposite to the 
direction of the plate movement, represents spray resistance. 

Let us consider the flow of a parallel stream of velocity V onto 
a stationary plate. Consider the depth of the fluid limited, equal to 
h. The influence of gravity on the flow of the fluid will be neglected. 

Flow of the fluid in such a motion is steady. Applying Bernoulli's 
equation to the particles on the free surface (Fig. 135)} and taking 
into account the fact that pressure on the free surface is constant, 
it can be shown that the velocity along the surface of the jet is con- 
stant also, and is equal to V. 

In the actual motion of the plate, the velocity in the jet is 
double that of the plate motion. Isolate by means of a control surface 
the closed volume of the fluid ABCDEF. ' According to the law of 
momenta, the flow of momenta through the control surface is equal to 
the vector of external forces applied to the volume, ABCDEF, 

The difference of momenta flow through the sections of the con- 
trol surface, AB and CD, is equal to VQ, where Q is the mass flow in 
the spray jet. 

Projecting the flow of momenta through the control surface ABCDEF 
and the vector of forces applied to the plate on the axis OX, we get 

\ = pVQ (l+cosO(j) 
and the total reaction is directed normal to the plate and is equal to 

R . 3L- = fVQ ^^^ = pVQ ctg^ 
sinot V sin 06 \ 2 



(1) See Figc li| 



148. 



These formulae are valid for any value of the angle o^; the theory 
discussed can be considered therefore as the non-linear theory of 
planing. 

As the fluid velocity in the spray jet in the case of a stationary 
plate is equal to V, the volumetric flow is 

Q = V^ 
where ^ is the thickness of the sprayo 

The force, R , in the case considered is equal to the spray re- 
sistance, R^ of the plate. Finally, the formulae for computation of 
the spray resistance and total hydroc^namic reaction in planing on the 
surface of a gravity-less fluid of arbitrary length at any angle of 
trim will take the following form: 

H ' P^^^ (I+COS06) (136) 

R=^V^^ctgf (137) 

If the plate of infinite span planes on the surface of a fluid, 
located in the field of gravity, waves are formed in its wake. In 
this case, the additional force of wave resistance will act on the 
plate. 

In the case of the motion of a planing plate on the surface of 
a perfect fluid, therefore, the total resistance is the sum of spray- 
making resistance, Rr, and of the wave resistance, Rg, Considering 
formula (135 )^ in the light of the above derivationj we find that 
Rg + R c = D tano^« 

(1) \ = D tanod+ R^p (135) 

D = total lift, ic eo dynamic + Archimedian 

R. = the force of friction 
tp 



1^9. 



The sum of forces Rg + Rr represents the projection of the re- 
sultant force of normal pressures, distributed along the plate, onto 
the direction of motion, i.e. represents resistance due to pressures. 

With the growt.h of relative velocity, the part played by the 
gravity forces diminishes in comparison with inertial forces; there- 
fore the part played by wavemaking resistance in the total resistance 
of the plate diminishes. 

The comparison of the relative importance of the wavemaking and 

spray resistance in the total resistance of the planing of the flat 

(1) 
plate is given on Figure I360 ^ 

The complicated calculations permitted L, I. Sedov to obtain the 
formula for computation of the ordinates of the free fluid surface, 
hydrodynamic pressure, lifting force, resistance and the hydrodynamic 
mcment acting on the planing surface. 

In eases of large relative velocities (small values of the para- 
meter V= 2 Y ^ ^^ ^^^ formula for calculation of the lifting force, re- 
sistance, and hydrodynamic moment of the flat plate were obtained by 
means of Sedov' s theory; 

R = R^ =pTT(i -TTv^ - l494-)a M^oC (I38) 

R^ = Rod (139) 

M = 2r (1 „ §1^) a^vl^ (11,0) 

where a = tt and F = V/\/gt. 

These formulae show that at the large relative velocities and 
small angles,©^ 5 the lifting force and hydrodynamic mcment in planing 

(1) See Figo Hi 



so. 



are linear functions of the angle of trim. 

More accurate numerical calculations of the lifting force and 
moment by Sedov' s theory were made by Uo S, Chaplygin for a large 
variation in the parameter ,V • Results of these calculations are 
compared on Figo 137 with results computed by formulae (I38) and 
(IhO). On Fig. 137(a) and (b) are plotted the dependence of C,/cC 
and the relative distance of the center of pressure from the rear 
edge of the plate '^^/J? on the number Fr« = v/VgJ?. Ccanparison of the 
curves shows that beginning with Fr, = 2<.8, i.e. withV= 0.06iij the 
results of the refined numerical computations and of calculations by 
approximate formula coincide« In the case of very large Froude 
numbers, Fr., it can be assumed that>^= 0; this assumption means 
physically that the gravity of the fluid is not considered. With 
the introduction of such an assumption, the formulae for the flat plate 
are further simplified to: 

R = a^ = pJTa V^ (liil) 

R^ = pTia V«^ (lii2): 

M = |pTra^V?>^ (IU3) 



Were the distance of the center of pressure from the trailing 
edge of the plate computed by means of the above formula for the 
moment, it would be found that jK , = (3/li)2a (i.e. m,^ = ol^)°, this 
agrees well with test results. 

Let us compare formula (lijl) and (lii2) with the formulae for the 
computation of corresponding quantities obtained in the wing theory 
for a flat plate of infinite span in a streamline flow at small angles 

(1) See Fig. lit 



51. 



of attacks 

R^ = 2pTra y^c6 (llili) 

M =^77'a^V^od (lli5) 

The flat plate represents the simplest wing. The comparison of these 
formulae shows that in planing at large Froude numbers the lifting 
force and moment of the plate is half that obtained in the flow 
around the plate deeply submerged in a fluid. This relationship 
points to the analogy of the flow on the lower part of the plate 
in both cases. 

The development of the analogy between the planing surface and 
the wing has permitted application of a number of relationships ob- 
tained in wing theory to stucfy the phenomena of planing. 

From the theory of the wing it is known that in a flow about a 
plate at an angle of attack 0^5 in accordance with the theoran of 
Joiikowskyj a lifting force occurs j normal to the direction of the 
flow at infinity o The velocity at the trailing edge of the plate 
has a finite value, and the streamlines flow off smoothly. At the 
sharp leading edge the velocity tends to become inf inite^ and in 
connection with this 3 the pressure in the vicinity of the leading 
edge drops sharply, ttydrodynamic pressures are normal to the surface 
of the plate, and their resultant, P, is also directed along the 
normal to the plate (Fig. 138,a),^ ' 3h addition to the force P, a 
suction force, S, acts in a direction along the plate. The total 
hydrodynamic reaction^, the lifting force of Joukowsky, R, is the 
vector sum of forces P and S; 

(1) See Fig. 1^ 



52. 



R = P + S 

The analogy of the flow in the space under the place in the 
case of planing at small angles of attack, and in the case of the 
plate fully immersed in a fluid, is valid from the trailing edge to 
close proximity of the leading edge. Analogous also are pressure 
distributions in this part of both flows. 

The flow is different only in the region of close proximity to 
the leading edge where the spray jet directed forward occurs along the 
plate (Fig. laajb).^-"-^ 

From the above reasoning, the results of the study of flow about 
a plate fully immersed in a fluid can be used for the study of flow 
under the planing surface if the flow about the leading edge (where 
velocities grow to infinity) is replaced by streamline flow reproduc- 
ing the spray jet. Such a replacement at small angles of trim will 
cause but an immaterial change in the area of the noraml pressure plot 
diagram at the lower edge of the plate. Taking into account the fact 
that the flow in planing occupies only the lower part of the space, it 
can be expected that the resultant of the normal force R at the planing 
surface is equal toi 

R =7/2. 
The validity of this relationship was already noted earlier. 

Developing the analogy, we can suppose that the reaction of the 
spray jet is directed in the opposite sense to the force of suction S, 
and in magnitude is equal to R, = 3/2. Making use of formula (I36) 
for the spray resistance in the case of planing on the surface of a 

(1) See Fig. lU 



^3. 



gravity-less fluid 

Rj^ = Spv (1+cosC^)^ 
and letting cosOd= 1 for small angles,06 and TL = S/2, we can compute 
the thickness of the spray jet: 

where S = suction force of the wing and is given as: 

S = 2?7pa Vot^ 

The validity of the above analogy is conf inned experimentally at 
large Froude numbers up to values of the angle, 0^, from 7 to 10 de- 
grees. 

In the case of flat plate planing at an arbitrary angle of trim, 
the resultant of hydrodynamic forces, R, can be calculated by 
foiTOula (137) 

R = pV^J ctg ^ 

The curve K = f(pOs showing the ratio of the magnitude of force R, 
calculated by formula (137) to the magnitudes calculated by formula (llil), 
(i.ea on assumption of small angles 0^)j is plotted on Fig, 139<« 
Fig. 139 shows that with increase of the angles^, the magnitxide of 
the resultant hydrodynamic pressure, computed by non-linear theory, 
becomes smaller than that computed by linear theory, i.e. the analogy 
between the planing s-arface and a wing is disturbed. 

The theory of planing permits one to appraise the effect of 
shallow water and of limited channel width on the hydrodynamic charac- 
teristics of planing surfaces. 

(1) See Fig. li^ 



^h. 



The detailed investigation of the effects of shallow water on 
planing upon a gravity-less fluid at various angles of trim was made 
by U. S. Chaplygin. 

As a result of these investigations it is established that, in 
planing on shallow water at small angles of attack, the analogy with 
the wing moving between parallel walls is observed. Theoretical cal- 
culations show that hydrodynamic characteristics in motion on shallow 
water depend on the ratiOytrh, and the angleo^, where h is the water 
depth. 

With increase of the ratio j2:h, i.e. with a decrease of water 
depth, the lifting force at constant velocity increases. 

On Fig. liiO, ^ ^ curves are given permitting one to estimate the 
increase of the lifting forces caused by shallowness of water at various 
values of the parameter X/:h and angles of trim,0<^. 

Taking into consideration the effect of channel walls on planing 
without consideration of gravity indicates that, at the ratio of 
channel width to the width of the plate b:B <.7, hydrocfynamic forces 
acting on the plate increase^ at bsB^- 7 the effect of walls leads to 
a small (less than 2%) reduction of forces as compared to motion on 
the surface of unbounded fluid. 



(1) See Fig. Ik 



55, 



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56, 



TABLE II 
Compiutation of C. Using m Based on Geometric Angle of TrinijT 



^L ■ 


2/?r) + 2 - A 
2 


V 


3(1: + 


2/it) * 2 - aT^ 


1 
2(A - BZ) 




- 


2 J 




1 2 


3 

B(r+2/fr)-^2-A 

2 


k 

Of 


5 

2(A-Bi:) 


6 


7 


X Z' 


V®*©' 


^L 


).2 2 


3.683 


13.561I 


-.557 


3.606 


.077 


h 


3.871 


111. 985 


-1,131 


3.722 


.lli9 


6 


It. 061; 


16.516 


-1.723 


3.81i6 


.218 


9 


ii.360 


19.010 


-2.61ili 


a.0li5 


.315 


12 


ii.665 


21.762 


-3.602 


h.26l 


.ItOii 


18 


5.301 


28.101 


-5.633 


l;.7ii3 


.558 


21^ 


5.968 


35.617 


-7.803 


5.27li 


,69k 


30 


6.661 


lilj.369 


-10 100 


5.853 


.808 


).li 2 


2.365 


5.593 


-,283 


2.30I1 


.061 


k 


2Mh 


6.170 


-.583 


2.363 


.121 


6 


2.608 


6,802 


-o901 


2.ii29 


.179 


9 


2.801 


7.81i6 


-l.ii30 


2.537 


.26I1 


12 


3.00ii 


9.02li 


-1.958 


2.658 


.3I16 


18 


3.ii3ii 


11,792 


-3.165 


2.937 


.lt97 


2k 


3.887 


15.109 


-li.500 


3.256 


.631 


30 


ii.383 


19.210 


-5.988 


3.635 


.7li8 


).6 2 


1.925 


3.706 


-.191 


1.875 


.050 


li 


2.022 


li,088 


-.hoo 


1,920 


.102 


6 


2.123 


li,507 


-.627 


1.970 


.153 


9 


2.282 


5.208 


-•>999 


2.052 


.230 


12 


2.I150 


6,003 


-l.lilO 


3.II12 


.308 


18 


2.812 


7.907 


-2.3ii3 


2.359 


.li53 


2k 


3.210 


10.30I1 


-3.1il6 


2.62li 


.586 


30 


3.62U 


13.133 


.-1;.617 


2.918 


.706 


).8 2 


1.705 


2e907 


-,ll;l 


1.662 


.01^3 


k 


1.791 


3.208 


".309 


1.703 


.088 


6 


1,880 


3.5311 


-,li90 


1.7U5 


.135 


9 


2.022 


li,088 


-.793 


1.815 


,207 


12 


2,173 


1^.722 


-1.135 


1.893 


.280 


18 


2,501 


6.255 


-1.932 


2.079 


.li22 


2k 


2.860 


8.180 


-2,868 


2.305 


.555 


30 


3.2iili 


10,52li 


-3.931 


2.568 


.676 



57. 



1.0 



2.0 



k.o 



6.0 



8.0 



TABLE II (Cont'd.) 



2 


3 

B(j;+2/7r)+2-A 
2 


li 

Of 


5 

2(A-Br) 


6 


7 


r 


\p^d 


^L 


2 


1.57ii 


2,ii77 


-.118 


1.536 


.038 


h 


1.6$2 


2,729 


-.25I1 


1.573 


.079 


6 


1.735 


3.010 


-.1(07 


1.612 


.123 


9 


1.866 


3.1i82 


-.670 


1.677 


.189 


12 


2.007 


a. 028 


-.971 


1.7^8 


.259 


18 


2.31ii 


5.355 


-1.685 


I.9I6 


.398 


2lt 


2.6^2 


7.033 


-2.539 


2.120 


.532 


30 


3.016 


9.096 


-3.520 


2.361 


.655 


2 


1.310 


1.716 


-<,o6U 


1.285 


.025 


h 


1.375 


1.891 


-.lii5 


1.321 


.o5h 


6 


lo ilUii 


2.085 


-.2ii3 


1.357 


.087 


9 


1.555 


2.i4l8 


-.1^23 


1.1^12 


.lli3 


12 


1.675 


2.806 


-.61^2 


l.ii71 


.20I4 


18 


1.9iil 


3.767 


-1.191 


1.605 


.336 


2i4 


2,238 


5.009 


-1.881 


1,768 


.li70 


30 


2.561 


6.559 


-2.698 


1.965 


.596 


2 


1.178 


1.388 


-,036 


1.163 


.015 


it 


1.236 


1,528 


-.090 


1.199 


.037 


6 


lo298 


1,685 


-.161 


1.23ii 


.061i 


9 


1.399 


1.957 


-.300 


1.287 


.112 


12 


1.509 


2,277 


-.ii78 


1.31a 


.168 


18 


1.751i 


3.077 


-,9kh 


1.1^60 


.29li 


2ii 


2,031 


I4.125 


-1.552 


I.60I1 


.li27 


30 


2.333 


5.to 


-2.287 


1.776 


.557 


2 


I.I3I1 


1,286 


-.027 


1.122 


.012 


h 


1.190 


l.lil6 


-.071 


1.160 


.030 


6 


1.250 


1.563 


-.133 


1.196 


.05I1 


9 


I.3I47 


l,8lii 


-.259 


1.2li7 


.100 


12 


l.i;53 


2,111 


-.ii23 


1.299 


.151^ 


18 


1.692 


2.863 


-.862 


l.lilli 


.278 


2li 


1.961 


3.8ii6 


-l,iii42 


1.550 


.liii 


30 


2.257 


5.09ii 


-2.150 


1.715 


.51^2 


2 


1.112 


1.236 


-.022 


1.102 


.010 


li 


1.167 


1.362 


-.062 


l.liiO 


.027 


6 


1.225 


1.501 


-.120 


1.175 


.050 


9 


1.321 


1.7li5 


-.238 


1.227 


.0911 


12 


1,1|26 


2.033 


-.395 


1.280 


.lii6 


18 


1,661 


2.759 


-.821 


1.392 


.269 


2li 


1.927 


3.713 


-1.387 


1.525 


.li02 


30 


2,219 


h>92k 


-2.081 


1.686 


.533 



TABLE II (Cont'd.) 



58, 



1 


2 


3 

B(t+2/?r)+2-A 
2 


h 


5 

2(A-Btr) 


6 


7 


X 


V®*©' 


^L 


.0 


2 


1.099 


1.208 


-.020 


1.089 


.010 




h 


1.153 


1.329 


-.057 


1.128 


.025 




6 


1.211 


I.I166 


-.111 


I.I6I1 


.Oli7 




9 


1.305 


1.703 


-.226 


1.215 


.090 




12 


l.li09 


1.985 


-.379 


1.267 


.lii2 




18 


1.6ii2 


2.696 


-.796 


1.378 


.26ii 




2ii 


1.906 


3.633 


-1.355 


1.509 


.397 




30 


2.196 


a, 822 


-2.0U0 


1.668 


.528 



59, 



TABLE III 

Computation of Lift Coefficient Usinp a Virtual Mass, m 
Based on Effective Anf le of Attack, Qt 



X 



'=L= 


4- (2.292 - - 


^)r-3. 


''' - 41 'l' 




"2 


.4*(^f' 


- 4.584) t 


6.283r 





2 


3 


4 


5 


6 


r 

deg 


Coeff of 


Coeff of 


Const 


Solution 




P 


q 


r 


^L 


2 


-3„354 


6.059 


-.219 


,036 


4 


-3„329 


6.119 


-.439 


.074 


6 


-3 .304 


6.178 


-.658 


.113 


9 


-3.266 


6.267 


-.987 


.166 


12 


-3.228 


6.356 


-1.316 


.232 


18 


-3.152 


6,534 


-1.974 


,350 


24 


-3.077 


6.712 


-2.631 


.482 


30 


-3 .002 


6.889 


-3.288 


.610 


2 


-2.826 


4.950 


-.110 


,022 


4 


-2.774 


4.900 


-.219 


.045 


6 


-2.721 


4,850 


-.329 


,070 


9 


-2.642 


4.774 


-.494 


.110 


12 


-2.563 


4.698 


-.658 


.151 


18 


-2 .406 


4.547 


-.987 


,250 


24 


-2.248 


4.397 


-1.316 


,360 


30 


-2 .090 


4.245 


-1,644 


.480 


2 


-2.563 


4.395 


- ,055.. 


.012 


4 


-2.496 


4.290 


-.110 


.026 


6 


-2 ,,430 


4.185 


-.164 


.040 


9 


-2.331 


4.02 7 


-.247 


.064 


12 


-2.231 


3.870 


-.329 


,090 


18 


-2.032 


3.553 


-.494 


.155 


24 


-1.833 


3.238 


-.658 


.237 


30 


-1.635 


2.923 


-.822 


.330 


2 


-2.475 


4,210 


-.03 7 


.009 


4 


-2 .404 


4.086 


-,073 


.017 


6 


-2.333 


3.963 


-.110 


,028 


9 


-2.227 


3.777 


-.164 


.044 


12 


-2.121 


3.592 


-.219 


.060 


18 


-1.908 


3,222 


-.329 


,110 


24 


-1.696 


2.852 


-.438 


.170 


30 


-1.484 


2.481 


-.548 


.240 



60. 



1 

K 



10 



TABLE nK Cont'd.) 



i 


3 


4 


5 


6 


r 

deg 


CoefC of 


Coeff of 


Const 


Solution 




V 


^ 


r 


=L 


2 


-2,431 


4.118 


-.02 7 


.007 


4 


-2,358 


3,985 


-.055 


,014 


6 


-2,285 


3,852 


-,082 


.021 


9 


-2.175 


3,656 


-.123 


.033 


12 


-2,065 


3,454 


-,164 


.049 


18 


-1.845 


3,056 


-.247 


.085 


24 


-1,626 


2.659 


-.329 


.134 


30 


-1.407 


2.261 


-.411 


.200 


2 


-2 ,404 


4,062 


-,022 


.005 


4 


-2,330 


3,924 


-,044 


.011 


6 


-2,255 


3,786 


-.066 


.018 


9 


-2,144 


3,579 


-,099 


,029 


12 


-2,032 


3,372 


-,132 


.040 


18 


-1,808 


2,957 


-,197 


,069 


24 


-1,585 


2,543 


-,263 


olio 


SO 


-1.362 


2,129 


-.329 


.160 






<;-. 
<« 







»o 
















^O 






X 






§ 




1^ 






-P 












-p 


•fi 






o 




c- 




CO 


^ 






(D 




o 




p 








■ — ■ 




CO 




-3 

1 


C\) 






>> >> 


^--v 


lO 




1 

1 


ID 




>> 


ca OS 


T) 






o 






CO 


tn U 


a> 


-p 




^ 


T) 




u, 


&, (O, 


0) 


00 




o 


00 




Oi 


U CO 


(X 
(0 


1 




in 


V 

u. 




10 


Cm <m 




a 




-p 






Cm 


O O 


•H 


o 

+3 




CO 


to 




o 


t< U 




(0 


f2 


CX 


(D 




i-. 


OS CO 


^ 




3 


o 


>H 


CO 


CO 


a> a> 


o 


(D 


u 


+> 




o 


0) 


i-i i-H 


•H 


> 




CO 


C~ 


LO 


rH 


o o 


-P 


Ob 


-p 




P. 




O 




CO 


(D 


« 


a> 


o 


P 






I-I 


s: 


<D 


> 


p 


s: 




./^ (0 


I-H 




fi 


00 


CO 




to 


;3 3 


•H 


<M 




0) 




£^ 


3 


-p +> 


O 


<iH 


© 


-C 


iVh 


o 


-P 


OS cs 


CO 


o 


(D 




<Vh 


-p 


CO 


t. t< 


o 




ta 


-P 


o 


CO 


Jh 


03 OS 




-p 




4) 






CO 


PL, &, 




(0 




(n 






(X, 


P. D. 




3 




0) 






iX 


< < 




•-3 




« 






< 


>- C- 


t- >- O O- lO 




t- 






o 


1— 1 I-H 


C~- C^ C^ CD C\J 




to 






CO 



o 

o 

-p 
«s 

10 

o 

-p 



o 

00 
o 

o 

a 
o 
-p 

CO 

e 

o 
-p 
-p 
o 

CQ 





i4 




8 




iH 




^ 








•H 




TJ 


^ 


8 


o 


r-t 


o 




I-H 


S 


^ 


o 


W; 


>> 


c 


CO 


•H 


Ih 


T) 


P4 


8 


to 


,— 1 


<D 




e 


G 


o 


O 


CO 




V«j^ 


>> 




(X, 




u 


c- 


ix 


tt 


CO 


o 




+3 


(U 


CO 


e 




o 

CO 


8 



61 



<M CM CO to ■^ 



'* 



CJOtOOtOtnyCDLDOOlDOOOtomO 
t~-CV3C>JcoCMC£)tD(X>C0O)CDcO^tOcO 



Eh 





« 








O 


O 


o o 


CO o o 


o 




> 


a 






r— 1 


rH 


c^ c^ 


CD CD r-H 


CO 




<5 


3 









° 


O 


O O 


n 




« 


« 






W 


CM 


CO CO 


•^ ^ in 


in 




Uh 


<M 
© 






to 

o 

o 












T) 


















E-H 


© 


Jl;. 
















a3 


•H 


fl 
















M 


I-H 


■H 
















W 


&, nd 
















CO 


(X 


00 


fi 


to 


CO 


CO 


t^ >- 


CD CD CO 


CO 




<! 


3 


t. 


,0 


CO 


to 


c- c- 


in in '^t* 


^ 


< 




*i5 


rH 


o 


o 





O O 





e-i 


TJ 








cvj 


CVJ 


CJi O) 


CO CO CD 


CD 


«a5 


CO 


o 












rH nH tH 


iH 


Q 




p 

<M 


CO 

3 

-p 
















•H 


o 


00 


to 


CO 












-P 




u 


^ 


CO 












CO 


i 


a 


rH 


o 












-P 


Oi 




C30 












to 




a, 
< 
















c 




















o 


u 






I-H 


iH 


ID to 


■^ T^ iH 


iH 






© 


CO 




CO 


CO 


o o 


CO CO !>■ 


C>- 




•d 


+3 


^ 







o 


o o 


O o 


o 




03 


^ 


I-H 




o 


o 


CO CO 


iH iH -^ 


-* 




^ 


^ 






I-H 


rH 


r-t M 


CM CM CM 


CM 




g 


W) 


















•H 


© 


^3 




CX> 
1— 1 












Em 


o 


















TJ 


© 






cv} 




00 


CM 00 






© 


CO 






c- 




c- 


C-- C- 






© 


^\> 






S 


o £ 


2 


s: 




Cm-P 






c- 




c> 


[^ t> 






CO 


Cm 






CVI 




CM 


CM CM 






© 




















V. 




















3 









a> 


O 


nH CVJ 


CO -^ in 


CD 




-P 


o 






•<* 


lO 


in in 


in in in 


in 




o 


s^ 






1X3 


lO 


LO in 


in in in 


in 




•H 




















04 




















c 


„ 


















3 


o 






I-H 


CM 


to -^ 


in CD i>~ 


00 




« 


i=5 

















OootnocoincDincooininincoOLn 

cot— CMCMtOCMCDCDCMCMCMCMCOCD t-CD 
oooooooo 000*^00 eo 

rHrHT^'<;;t<LninCMCM'HfH'^'<:t*Ln'<;^CMCVl 



o 
o 



in 
CD 



CO CO CD CD CO 
in in CM CM •<;»< 



rH iH CM CM iH rH 



ooininmini— irHCMOinin 

^CDCOCDCD';t<'<;t<000'^'^ 



oooDcoincvjcno^ 

iHiHCNjCM COCOiHiH 



00 
CM 



00 



CO 
CO 



00 



rHiH'^'^i^CDCDcOCOCOCOaiasOOOCOCO 

C0C0ininC-C-CnCT5Cr5O)CDCDrHCMt-C~- 

ooooooooeoeooooo 

^^t-t-CMCMCT5O>00a0CDCDsi<O>-C- 
iHrHCMCMtOCOiHrHrHrHcOCO'<;f<^C0CM 



CM 



O 
CO 



CM 






CD CM 




CM x;^ CO 






CO O 


C- 






CD t- 




C- CO c- 






I> <T> 


• s 


S 


s: 


o o ^ 


s 


O O OS 


s 


£ 


o o 


CM 






CM CO 




C- I>- I>- 
CO CO CO 






CO CO 



t-000^OrHCMtO"5j<incDC-(X)O^O 
inLninCDCOCDCDCDCDCDCDCDCOC- 

ininininininininLnininininin 



I rH 

1 in 



C7)0'-iCMeO'!;t<incDC-oocnO'-<CM co';*^ 

rHrHrHiHrHiHiH iHiHrHCOCOCO COCO 



62 



TABLE V 



Wave Rise "Correction" Factors for Wetted Length Determinations 

from Heave Readings 

















b = 


2" 


Angle 


Photo 


Veloc ity 


Speed 


Wetted 


Heave 


h^ 


Wave Rise 


of 


Noo 


V 


Coeff. 


Length /Beam 


(Reading 


sinT^ 


Factor 


Trim 




(ft/sec. ) 


C 


Ratio 


Min\is 




(X-h^ sintr) 


IT 








V 


from Photo 


Ref.) 






(deg.) 










X 


K 








51i9 


27. 


.72 


11,96 


2,30 


1,17 


1.89 


.Ix^ 




5^0 


27. 


.72 


11,96 


2.30 


1.17 


1.89 


.hi 




^51 


27. 


.78 


11.99 


5.07 


2.77 


IM 


.58 


/% 


552 


27.78 


11,99 


5.05 


2.77 


ii.li9 


.56 


18° 


553 


27. 


.72 


11,96 


6.55 


3.70 


60 00 


.$^ 




55U 


27. 


.72 


11.96 


6.I13 


3.67 


5.95 


.I48 




555 


27.78 


11.99 


7.22 


a.23 


6.85 


.37 




556 


27.78 


11.99 


l.hh 


ii.37 


7.08 


.36 




557 


27. 


.72 


11.96 


2o68 


1,80 


2.22 


.1^6 




558 


27. 


.72 


11,96 


2.63 


1,78 


2.19 


oUi 




559 


27. 


.72 


11,96 


5.87 


ii.25 


5.23 


.e\x 




560 


27. 


.72 


11.96 


5,82 


U.20 


5.16 


.66 


2U° 


561 


27. 


.66 


11.9ii 


7.25 


5.33 


6.55 


.70 




562 


27. 


.72 


11.96 


7.12 


5.25 


6.1^5 


.67 




563 


27. 


.72 


11.96 


3.80 


2.66 


3.27 


.1^3 




56ii 


27. 


.72 


11,96 


3.80 


2.65 


3,26 


.5ii 




^e^ 


27.72 


11,96 


2.52 


1,88 


1,88 


,61 




566 


27.8ii 


12,02 


2ol42 


1.85 


1.85 


.57 




567 


27. 


.72 


11,96 


5o72 


ii,90 


Iio90 


.82 


30° 


568 


27. 


.72 


11,96 


5.67 


li,90 


ii,90 


.77 




%9 


27. 


.72 


11,96 


7.03 


6,30 


6,30 


.73 




570 


27< 


.72 


11,96 


6.37 


5.63 


5.63 


.7ii 




571 


27. 


.90 


12, OU 


li,00 


3.30 


3.30 


.70 



63 



TABLE VI 
Reduction of Experimental Data, Correlating Cj- and X 

= -9^ 1^ b2 = .02777 ft.= q = l^V^ SiH^t: 

sec. 



^L = q b^ X 

Pict. V C^ q X Xb^ qXb^ L Cj^ 

5U9 27o72 11.96 7ii5.35 2.30 .06387 1;7.6055 10.61 .223 

5^0 27.72 11.96 7ii5<.35 2.30 .06387 I47.6055 10.61 .223 

^51 27.78 11.99 7ii8.58 5.07 .II1O8 105. iiO 18.05 .171 

552 27.78 11.99 7U8.58 5.05 .lii02 IOI1.95 18.05 .172 

553 27.72 11.96 7li5.35 6.55 0I819 135.58 21.8a .161 
55ii 27.72 11.96 7li5.35 6,li3 .1786 133.12 21.81^ .l6ii 
9?"^ "^l^l^ 11.99 71^8,58 7.22 .2005 150.09 2ii.71 .165 

556 27.78 11,99 7ii8.58 7.1iii .2066 15I4.66 2ii.71 .160 

557 27.72 11.96 7^5.35 2.68 .07i;i42 55.ii7 lii.8l .267 

558 27.72 11,96 7ii5.35 2.63 ,073014 51^.14; li4.8l .272 

559 27.72 11.96 7ii5o35 5.87 .1630 121. I49 27.5ii .227 

560 27.72 11,96 7ii5.35 5.82 .1616 120. ii5 27.51; .229 

561 27.66 11, 9U 7h2,13 7.25 .2013 lh9.39 32.76 .219 

562 27.72 11,96 7i45.35 7.12 .1977 II17.36 32.76 .222 

563 27.72 11,96 7li5.35 3o8o ,1055 78.63 19.93 .253 
561^ 27,72 11.96 7ii5.35 3. 80 .1055 78,63 19.93 .253 

565 27.72 11.96 7li5o35 2,52 .06998 52.16 18„93 .363 

566 27. 81^ 12,02 751.82 2oii2 ,06720 50.52 18.93 .375 

567 27.72 llo96 7)45.35 5.72 ,1588 118.36 36.69 .310 

568 27,72 llo96 7i45.35 5.67 .1575 117.39 36.69 .313 

569 27.72 11,96 7ii5.35 7.03 .1952 1145. ii9 l4i4.10 ,303 

570 27.72 11.96 7li5.35 6,37 0I769 131.85 liO„28 ,305 

571 27.90 12.0I4 755.06 I4.00 ollll 83,88 27.73 .331 



64 



TABLE VII 

Tabulation of Computed and Experimental Results for 
the Gravity Independent Lift Coefficient 
125 4 56 78 









Definit: 


ion of 


Definition of 








ift Based 


on V 


A Based on c^ 
e 


X r c^ 


C 

L Exp 


c 

L Comp 


L Comp 
^L Exp 


L Comp 


L Comp 
L Exp 


,30 2 


° 21,44 


.031 


,067 


2,16 


,072 


2.32 


.40 


18,51 


,031 


,060 


1.935 


,065 


2.10 


.45 


24,61 


,031 


,058 


1,87 


.063 


2,03 


.62 


20,53 


,033 


,049 


1,485 


,052 


1.58 


.68 


20,74 


,029 


,046 


1,585 


,049 


1.69 


.70 


24,92 


,029 


,045 


1,55 


,048 


1,65 


,87 


14,55 


,023 


,041 


1,78 


,041 


1,78 


.87 


22.88 


,028 


,041 


1,46 


,041 


1,46 


1,17 


20,04 


,027 


,035 


1.30 


,033 


1.22 


1.57 


25,01 


,022 


,02 9 


1,32 


,027 


1,23 


2,54 


13,37 


,019 


,022 


1,16 


,020 


1,05 


3,00 


13,66 


,015 


,020 


1,33 


,018 


1,20 


3,00 


13,72 


,015 


,020 


1,33 


,018 


1,20 


3,00 


16,99 


,015 


,020 


1,33 


,018 


1,20 


3,00 


21,87 


,015 


,020 


1,33 


.018 


1,20 


4,79 


12,20 


,012 


,015 


1,25 


,012 


1,00 


6,50 


25,01 


,009 


,012 


1,33 


,008 


,89 


6,50 


18.64 


.010 


,012 


1,20 


,008 


,80 


7,79 


13,39 


,009 


,010 


1,11 


.005 


,555 


8,12 


23,39 


,009 


,010 


1,11 


,005 


,555 


8,27 


12.78 


,009 


,010 


1,11 


,005 


,555 


,25 4 


° 14,40 


,082 


,140 


1,71 


,150 


1,83 


.35 


19.98 


,091 


,125 


1,37 


,135 


1,48 


.38 


16,99 


,117 


,123 


1,05 


,133 


1,14 


,38 


24,86 


,091 


,123 


1,35 


,133 


1,45 


.50 


13,57 


,092 


,110 


1,20 


,120 


1,31 


.54 


13,68 


,084 


,108 


1,29 


,115 


1,37 


.62 


21,66 


,073 


,100 


1,37 


,105 


1,44 


,85 


24,92 


,073 


,085 


1,16 


,088 


1,20 


1,45 


21,87 


,055 


,063 


1,14 


.057 


1,04 


1.45 


12,81 


,054 


,063 


1,17 


,057 


1,06 


1,50 


16,26 


,054 


,062 


1,15 


,056 


1,04 


1,52 


16,23 


,053 


,061 


1,15 


,055 


1,04 


1,64 


16,37 


,048 


,059 


1,23 


,053 


1.10 


1,66 


12,86 


,049 


,059 


1,20 


,052 


1,06 


2,20 


24,80 


,041 


,051 


1,24 


,042 


1,02 


3,66 


10,97 


,029 


,039 


1,34 


,029 


1.00 


4,07 


24,77 


,029 


,036 


1,24 


.026 


.895 


4,54 


13,48 


,026 


,034 


1,31 


.024 


.925 


4,63 


20,95 


,027 


,034 


1,26 


,023 


,854 


4,66 


18,13 


,025 


,033 


1,32 


,023 


.920 


4,82 


17.51 


,026 


,033 


1,27 


,022 


,846 


5,14 


22,97 


,027 


,032 


1,18 


,021 


.778 


5,54 


17,57 


,022 


,031 


1,41 


.020 


,910 


7,16 


12,00 


,021 


,02 9 


1,38 


,020 


,955 


8,54 


15,46 


,019 


,026 


1.37 


.01.2 


,630 



65, 



TABLE VII( Cont'd.) 



5 6 7 8 
Definition of Definition of 









m Based on X 


m Based on Of 
e 


k 1 








c 

L Comp 




L Comp 


tr C^ 


L Exp 


L Comp 


L Exp 


L Comp 


c 

L Exp 


,22 6 


° 19.89 


.147 


,207 


1,41 


.208 


1.42 


.25 


24,77 


,139 


.205 


1.47 


205 


1.48 


.30 


16,68 


,153 


,194 


1,27 


.195 


1.27 


.38 


24.92 


.163 


,181 


1.11 


.184 


1.13 


.55 


12,69 


.143 


,158 


1.10 


.156 


1.09 


.55 


16.32 


,145 


,158 


1.09 


,156 


1.08 


,65 


21,90 


.123 


.150 


1.22 


,146 


1.19 


.66 


16.27 


,122 


,149 


1.22 


,145 


1,19 


,67 


25,10 


.131 


.148 


1.13 


.144 


1.10 


1.32 


12,96 


.096 


,107 


1.12 


.098 


1.02 


1.40 


24,46 


.086 


,105 


1,22 


.094 


1,09 


1,42 


24olO 


.088 


,104 


1.18 


.093 


1,06 


1,42 


17.60 


,087 


,104 


1.20 


.093 


1.07 


1.52 


20,86 


.084 


,100 


1.19 


.090 


1.07 


1,66 


17.37 


,076 


.095 


1,25 


,085 


1,12 


3,00 


17.60 


,060 


.073 


1,22 


,053 


.884 


3.05 


10.83 


.059 


,072 


1.22 


,052 


,880 


3.22 


24,16 


,057 


,070 


1.23 


,050 


,88 


3.32 


14,55 


.055 


,069 


1.25 


,049 


,89 


3.35 


19.86 


,055 


.069 


1.25 


,048 


,87 


3.41 


17.71 


.052 


.068 


1.31 


,047 


.90 


3.79 


10.72 


.049 


.065 


1.33 


,043 


.88 


5.20 


24.46 


,045 


.056 


1.24 


,031 


.69 


5.57 


20,74 


,044 


.055 


1,25 


.030 


.67 


5,62 


17.08 


.044 


,055 


1,25 


,029 


,66 


5,62 


12,41 


,044 


,055 


1,25 


.029 


,66 


5.75 


15.01 


.043 


.054 


1.26 


.028 


.65 


6.29 


17.81 


.039 


.053 


1.36 


.025 


.64 


7.29 


19.90 


,037 


,051 


1.38 


.021 


.57 


7,66 


16.27 


.036 


.051 


1,42 


,020 


.555 


7.79 


14.13 


,036 


,050 


1.39 


.020 


.555 


.91 9 


° 15.70 


,171 


.197 


1,15 


.180 


1.05 


,91 


11.83 


.167 


,197 


1,18 


.180 


1.08 


1.86 


18,28 


.117 


.146 


1,25 


.117 


1.00 


3.04 


11.80 


.091 


,121 


1,33 


.081 


.89 


3,10 


14.10 


.090 


.120 


1,33 


.080 


.89 


3,66 


15.40 


,084 


,114 


1,36 


.069 


,82 


5,41 


13.78 


,070 


.104 


1.49 


.048 


,69 


5,41 


11.98 


,071 


,104 


1.47 


.048 


,68 


7,16 


10,82 


.066 


.096 


1,45 


.037 


,56 


7.29 


12,30 


.066 


,096 


1,45 


.036 


.55 


.08 12 


° 24.98 


,256 


,455 


1.78 


,460 


1.80 


,10 


20,04 


,318 


.445 


1.40 


,445 


1.40 


.12 


16.96 


,371 


,435 


1.17 


.435 


1,17 


.12 


24.95 


,285 


.435 


1.53 


.435 


1,53 


.15 


21.59 


,305 


.425 


1.39 


.425 


1.39 



66, 



TABLE VII (Cont'd.) 









Definit 


ion of 


Definition of 








m Based 


L on "V 


m Based on 0( 
e 


X t c^ 


Ct i.^ 
L txp 


Q 

L Comp 


Q 

L Comp 
L Exp 


L Comp 


L Comp 
L Exp 


,20 12 


° 25,16 


,302 


„400 


1,33 


,410 


1,36 


.20 


12,75 


.393 


,400 


1.02 


.410 


1.04 


.25 


16.23 


.324 


,385 


1,19 


,390 


1.20 


,28 


21,59 


,294 


,380 


1,29 


o385 


1,31 


.30 


25.25 


.291 


,370 


1,27 


.375 


1.29 


.31 


24,98 


,198 


,367 


1.85 


,372 


1.88 


.40 


13.79 


„280 


,345 


1,23 


.345 


1,23 


.40 


17,32 


,318 


,345 


1,08 


.345 


1,08 


.45 


21.14 


.276 


,333 


1,21 


.335 


1.21 


,48 


24,40 


,254 


,328 


1,29 


.330 


1,30 


.53 


12,26 


,268 


,319 


1,19 


,315 


1.18 


.54 


12.22 


,264 


,318 


1,20 


.312 


1.18 


.70 


17,54 


,257 


,295 


1,15 


,275 


1.07 


.75 


24,40 


,238 


,288 


1,21 


,268 


1.12 


,75 


20,07 


,240 


,288 


1,20 


,268 


1,11 


.77 


10.80 


,236 


,285 


1,21 


,266 


1,13 


1.15 


24,49 


,204 


,248 


1,22 


,214 


1,05 


1.17 


20,89 


,209 


,247 


1,18 


,213 


1,01 


1.20 


14,95 


,206 


,245 


1,18 


,210 


1,02 


1,22 


12.38 


,204 


,242 


1,19 


,208 


1,02 


1.25 


12,50 


,195 


,240 


1,23 


,205 


1,05 


1.80 


24,28 


,164 


,210 


1,28 


,164 


1.00 


2.00 


23,27 


,161 


,204 


1,27 


,152 


,94 


2.05 


14,94 


,157 


,202 


1,29 


,150 


,96 


2.07 


21,01 


,154 


,201 


1,31 


,149 


,97 


2,62 


16,78 


,143 


,187 


1,31 


,125 


,87 


2,66 


12,22 


,140 


,185 


1,32 


,125 


,89 


2.90 


21,01 


,136 


.181 


1,33 


,116 


,85 


2.92 


11,74 


.138 


,180 


1,31 


,115 


,83 


2,97 


20.74 


,136 


.180 


1,32 


,114 


,84 


3,00 


16,23 


,134 


.179 


1,34 


,113 


,84 


3.07 


16.13 


,132 


,178 


1,35 


,112 


,85 


4,05 


16,78 


,123 


,167 


1,36 


,088 


,72 


4.07 


10,52 


,122 


,167 


1.37 


,087 


,71 


4,12 


18,67 


,121 


,166 


1,37 


,086 


,71 


4.20 


14,55 


,120 


,165 


1,37 


,085 


,71 


4.25 


12,02 


,118 


ol65 


1,40 


,085 


.72 


.12 18 


° 25,10 


.50Q 


,585 


1.15 


,580 


1,14 


,18 


12,69 


.441 


,5 60 


1,27 


,560 


1,27 


.18 


21,75 


,450 


,560 


1,24 


.560 


1,24 


.18 


25.50 


,473 


,560 


1,18 


,560 


1,18 


.20 


10,13 


,622 


,550 


.885 


,550 


,88 


,25 


21,26 


,490 


,535 


i,09 


,525 


1,07 


.26 


15,49 


,342 


,533 


1,56 


,523 


1,53 


.28 


10,13 


.444 


,529 


1,19 


,515 


1,16 


,28 


17,23 


,461 


,529 


1,14 


,515 


1,11 



67, 



TABLE VII (Cont'd,) 



7 8 

Definition of Definition of 









m Based on U 


m Based on CV 
e 


X tr c^ 


L Exp 


L Comp 


L Comp 
L Exp 


C^ 
L Comp 


L Comp 
^L Exp 


.28 18 


° 21.01 


.448 


.529 


1,18 


,515 


1.15 


.28 


24.43 


,433 


.529 


1.22 


,515 


1.19 


.35 


24.40 


.511 


.505 


,987 


.487 


.95 


.40 


17.63 


,446 


,492 


1,10 


,470 


1.05 


.40 


10.74 


.442 


,492 


1.11 


,470 


1.06 


.42 


10.83 


.432 


,488 


1.13 


.465 


1.07 


.44 


19.82 


.419 


,482 


1,15 


.460 


1.10 


,58 


15,01 


.424 


,456 


1.08 


,426 


1.01 


.58 


24,61 


,400 


.456 


1.14 


.425 


1.06 


.60 


20.83 


,409 


,450 


1.10 


.420 


1.03 


.65 


12,44 


,381 


,441 


1.16 


.416 


1.09 


.90 


23.18 


.361 


.408 


1,13 


.368 


1.02 


.92 


10.98 


.346 


,406 


1,17 


.364 


'1,05 


,95 


14.91 


,343 


,403 


1,17 


.361 


1.05 


1.02 


21.01 


.312 


.395 


1,26 


.360 


1.12 


1.35 


16.31 


.296 


.368 


1,26 


,310 


1.05 


1.38 


20.98 


,287 


.366 


1,27 


.307 


1.07 


1.47 


16.20 


.276 


.362 


1.31 


.299 


1,08 


1.50 


15.65 


.290 


.359 


1.24 


.294 


1.01 


1.55 


12.99 


,276 


,356 


1.29 


,289 


1,05 


1.87 


16.78 


,266 


,343 


1.29 


.261 


.98 


1.88 


14.61 


.265 


,343 


1,29 


.260 


.98 


1.98 


14.70 


,249 


.339 


1,36 


.254 


1,02 


2.02 


18.64 


,249 


.337 


1.35 


.250 


1.00 


2.02 


16.80 


.247 


.337 


1.36 


.250 


1.01 


2.02 


11,90 


.253 


.337 


1,33 


.250 


.99 


2,30 


11.96 


.223 


,328 


1,47 


,230 


1.03 


2,30 


11.96 


,223 


,328 


1.47 


,230 


1,03 


5.07 


11.99 


,171 


,284 


1,66 


.127 


.74 


5.05 


11.99 


.172 


.284 


1,65 


.128 


.74 


6.43 


11.96 


,164 


,275 


1.68 


.104 


,63 


6.55 


11.96 


.161 


.2 74 


1,70 


.102 


.63 


7,22 


11.99 


,165 


,272 


1,65 


,094 


.57 


7.44 


11.99 


.160 


.271 


1.69 


.091 


,57 


.22 24 


° 21.11 


.565 


.680 


1.20 


,686 


1,21 


.22 


17.35 


.579 


.680 


1.18 


,686 


1.19 


,24 


13,02 


.523 


.673 


1,29 


.675 


1.29 


.25 


io,ao 


,501 


,671 


1.34 


,675 


1,35 


.32 


17,45 


,568 


.645 


1,14 


,650 


1.16 


,32 


20,04 


,563 


,645 


1,15 


.650 


1,15 


,32 


24.40 


.558 


,645 


1.16 


,650 


1,17 


.38 


24,34 


.473 


,635 


1,34 


.636 


1.35 


.38 


10.80 


.480 


.635 


1,32 


,636 


1.33 


.44 


24.61 


.527 


.618 


1,17 


.615 


1.17 


.45 


20,86 


.544 


,617 


1.13 


,614 


1,13 


.49 


12.47 


,503 


.610 


1,21 


,608 


1,21 


,50 


15.01 


,492 


,605 


1,23 


.604 


1.23 



68, 



TABLE ViKCont'd J 









Definition of 


Definit 


ion of 








A Based on 


m Based 


L on OC 
e 


A X 








L Comp 




Q 

L Comp 


'y 


^L Exp 


L Comp 


^L Exp 


Q 

L Comp 


"^L Exp 


,55 24 


° 24,95 


.510 


.594 


1,16 


.585 


1.15 


.55 


20.77 


.448 


,594 


1.32 


.585 


1.30 


.55 


12.41 


,453 


,594 


1.31 


,585 


2.31 


,65 


18.39 


.484 


.578 


1.20 


,560 


1.16 


,70 


14,88 


,467 


.570 


1.22 


,550 


1.08 


.70 


20,92 


,459 


.570 


1.24 


,550 


1.20 


.70 


23.33 


,458 


.570 


1.25 


.550 


1.20 


,94 


11.65 


.434 


.539 


1.24 


,500 


1.15 


.95 


13.42 


,423 


.539 


1.27 


.498 


1,18 


.98 


20.62 


,419 


.536 


1.28 


,495 


1,18 


1.00 


16.16 


,408 


.533 


1,31 


.487 


1,19 


1.17 


16.59 


,437 


,517 


1.18 


,457 


1.05 


1.22 


18.54 


,416 


,514 


1.23 


,450 


1.08 


1.22 


11.90 


.419 


,514 


1.23 


,450 


1.07 


1,27 




.399 


.510 


1.28 


,443 


1.11 


2.63 


11,96 


,2 72 


.452 


1.66 


,308 


1.13 


2.68 


llo96 


.267 


,451 


l.,B9 


,306 


1.15 


3.80 


11,96 


.253 


,430 


1.70 


.245 


.97 


5.82 


11.96 


.229 


,412 


1.80 


,175 


.77 


5.87 


11.96 


,227 


,412 


1.81 


,174 


.77 


7,12 


11,96 


,222 


.406 


1.83 


,148 


,67 


7.25 


11.94 


,219 


,405 


1.85 


,146 


.67 


.11 30 


° 21.72 


,739 


.826 


1.12 


,835 


1.13 


,11 


16.41 


,719 


.826 


1.15 


,835 


1.16 


.11 


12.78 


.711 


,826 


1.16 


,835 


1.17 


.12 


24.98 


.740 


,822 


1.11 


.830 


1.12' 


.16 


20.86 


,796 


,814 


1.02 


,820 


1.03 


.16 


17.63 


.771 


,814 


1.05 


,820 


1.06 


.18 


10.13 


,692 


.807 


1.17 


.816 


1.18 


.27 


17.45 


.674 


.783 


1.16 


.785 


1,16 


.27 


10.89 


,665 


,783 


1.18 


.785 


1.18 


o31 


20,10 


,578 


,770 


1.33 


.765 


1.32 


,40 


12.41 


.622 


.749 


1.20 


.749 


1.20 


.40 


14.91 


.623 


.749 


1.20 


,749 


1.20 


.54 


14,98 


.598 


,"^18 


1.20 


.711 


1.19 


.73 


13.24 


,566 


.686 


1.21 


,668 


1.18 


.98 


11.99 


,514 


,658 


1.28 


.6^2 


1.21 


.98 


14.49 


.518 


.658 


1.27 


,622 


1.20 


2.42 


12.02 


.375 


.583 


1.55 


.443 


1.18 


2,52 


11.96 


.363 


.581 


1.60 


.435 


1,20 


4.00 


12,04 


.331 


.557 


1.68 


,330 


1,06 


5,67 


11.96 


,313 


.544 


1.73 


,253 


.81 


5,72 


11.96 


,310 


.543 


1.75 


.252 


,81 


6.37 


11.96 


,305 


.540 


1.77 


.234 


.77 


7.03 


11.96 


.303 


,537 


1,77 


.217 


.72 



69 



O 00 Oi ^ CO to 
rH O r-H e-l CM CM 



CM CO O C^ 
Cr> r-{ 

to CO ^ to 



in o CO c^ to ^ 

cnas into cj>5* incj) 

to in c- CO CO «o 



< 
E-i 



Pi 
O 
•H 
-P 
OS 



o 
-p 



CM 

I 

-P 

Ms 



;^h'^ 



cr> 



o 



o 
-P 

s^ 
o 

•H 

.-p 
o 
v> 
u 
u 
o 
o 



CO 

o 

•H 
Vi 
•H 

r 



O CiS 

-p PQ 

6 

o <u 
o o 

^^ C 

o o 

•H 
P 
•H 

•H 
<Vh 
(D 
Q 



OS 
CD 

in 
to 



CO 

p 
I 

in 

CO 



CO 



<^ ^ 



1J 
a> 
•p 
o 

o 
o 

>^ 
o 



CD 



in 



Kj< 



to 



CM 



in 


o 


o 


r— t 


CJl 


o 


o 


i-H 


1— 1 


CM 


■-I 


■* 


o 


o 


o 


O 


o 


o 



O CM O to O CD 
CO CO •«*< CO C-- Ti* 
O O O O O --H 











in 




















in 


o 


r-l 


CM 


o 


o 


CM CM 


-^ 


CD 


t- 


iH 


O 05 


00 


t- 


o 


.— 1 


(-) 


CM 


CM 


"* 


CO CD 


•^ 


CO 


c- 


in 


CM O 


C-- 


C- 


o 


o 


c 


o 


O 


o 


o o 


o 


O 


o 


r— 1 


<H CM 


r-i 


CM 



in 

in o to ^ CM o 

O <-l rH CM CM ^ 

o o o o o o 



^ 


CM 


en 


CD 


in 


in 


^t* 


03 


CD 


CO 


CO 


CO 


^ 


CO 


CO 


in 


CO 


O 


cr> 


CO 


o 


o 


o 


o 


o 


rH 


<-l 


CM 


rH 


CM 



CD 


o 


in in 


■<;*< 


o 


CO 


'vt^ 


in 


CM 


CD i-H 


o t^ 


CO 


o 


O 


f-t 


<H CM 


CM 


'^ 


CO 


CD 


in 


o> 


O) CD 


in CM 


rH 


o 


O 


o 


o o 


o 


o 


o 


o 


o 


o 


o >-* 


I-H CM 


CM 


CO 



t> r-i 


C^ 


CO 


CO 


O 


^ 


c- 


CM 


in 


o 


00 


O 


tr~ 


>* 


O 


O r-i 


rH 


CM 


CM 


■<^ 


'^ 


co 


CO 


en 


rH 


CD 


C-- 


CO 


'^ 


I-H 


O O 


O 


O 


O 


o 


o 


o 


O 


o 


•-i 


t-i 


r-i 


CM 


CM 


CO 



O '^ 


r-^ 


(T> 


CO 


CD 


rH rH 


>;»< 


CM 


o 


o 


CO 


CM 


00 


-St* 


r-\ r-i 


CM 


CM 


CO 


•^ 


in t>- 


c^ 


o 


to 


CO 


C7) 


in 


c^ 


CM 


O O 


O 


o 


o 


o 


o o 


o 


r-i 


M 


r-i 


r-i 


CM 


CM 


CO 



to CD 


CO 


CM 


o o 


^ o 


O CM 


in 


to 


in 


r-i 


O CM 


r-i r-i 


CM 


to 


^ in 


CD CO 


CD rH 


in 


a> 


CO 


t> 


CO in 


O O 


o 


O 


o o 


O O 


O rH 


rH 


r-i 


CM 


CM 


CO CO 



CD CO 


CO c^ 


o 


CD 


CM rH 


•^ 


C^ 


CM 


^ 


in 


CD 


CO t~ 


r-i r-i 


CO CO 


in 


in 


CO O) 


r-i 


CM 


CD 


<H 


CO 


CJ> 


05 00 


O O 


o o 


O 


O 


o o 


r-i 


r-i 


r-i 


CM 


CM 


CM 


CO CO 



CM CM 


in 


in 


r-i 


r-i 


r-i r-i 


CM CM 


o 


o 


00 


CD 


CM in 


CM CM 


•^ 


'^ 


C^ 


t>- 


r-i r-i 


in in 


in 


in 


in 


CO 


CO CM 


O O 


o 


o 


o 


o 


r-i r-i 


r-i r-i 


CM 


CM 


CO 


CO 


■^ --e* 



CD CM 


in 


t> 


CO 


r-i 


O 


CO 


O 


CD 


CO 


c- 


>- 


a> 


CO 


r-i 


CO CO 


tv 


CD 


rH 


o 


C- 


in 


CO 


o 


in 


r-i 


00 


r-i 


r-i 


r-i 


O O 


o 


o 


r-i 


r-i 


r-i 


r-l 


CM 


CvJ 


CO 


CO 


•^ 


^ 


CO 


in 



0) 
-P 

o 
o 

o 



?^ 



^ 



-P 

3 

O 

o 



1-^ >-^ 
o o 



°^ 



'^ 



CO 



O) 



CO 



CM 



o 
to 



TABLE IX 
Values of the Lift Coefficient for Low Aspect Ratio Airfoils 



70. 



2TTU 



1 ■»■ 



2 XTT 

Jf* UX sin 



(Cy found from the momentum 
equation with m based on od) 



X = 2o86 



X = 7.16 






^L 





^L 




0( 


°L 





^L 


























10 


.Ul6 


10 


.291 


10 


.2iih 


10 


.165 


20 


.917 


20 


.699 


20 


.619 


20 


MS 


30 


l.ii82 


30 


1.193 


30 


1.087 


30 


.915 



Gl = 



2 ^2X ^^ 



IfoC 



4 



^ (jj: . 1) -HOC 



4. 



(C, found from the momentum 
equation with m based onoC^ 



X = 1 





oc 


^L 








10 


.381 


20 


,807 


30 


1.271 






c 


OC 


L 








10 


.238 


20 


.508 


30 


082I1 



X = 


2.86 



OC 


^L 








10 


0I78 


20 


o388 


30 


0637 



X = 7.U6 






G 


^ 


L 








10 


.076 


10 


.171 


30 


.290 



Schematic for Virtuol Mass Definitions 



Fig._L 






m based on r 



m bated on a^ 



Schematic for Spray Thickness, 
Stagnation Point, and Spray Root 



Fig 



l^aius 



Sprou R«ak 



Stagnation Rjint 



Undistiu'bcd Water 3uriac6 




Sireamli'nc 





.90 
.80 
.70 
.60 

.50 
.40 
.30 

.20 

■ 

.10 
. 
































i 1 1 1 ' ■ 1 1 
P.nmniiipH Wnliioc rkf Pi we a fminH 










OUIII^UICU VUIUOS Ul ^1 "'■ '^ IvUIIU 










from the Quodrotic 
C, =B(T + 2/7r)+2-A 










-V 


2 






Lift Coefficient, 0, 






'^rB(T + 2/Tr)+2-A|'+2(A-BT) 
L o J 






\ 


Fig._2- 
1 1 1 1 1 1 1 






























\ 


















-so* 










\ 


\ 
























"V. 






, 
















\ 


















-24» 




\\ 


\ 


\^ 




















l\ 


V 








^- 











-le* 








\\ 


\ 
























\\' 


\, 


^ 






















V 




^ 
















r 






V 


^^ 


^- 
















-e* 

-4* 






V 

















































c 

1 


1 i ' ' 


1 1 • 1 -^ 

6 8 10 

Length -Beom Ratio ,X , , 







.90 
.80 

.70 
60 

• 
































r 


Ann n II 


4aW VinliiA 


• A« 


n 


ua \ #^ii nH 










\j 


from the Cubic 
^+rj'2.292-«.»7l \^_ 1 -2.379lc.* 

"fe.284_ 4^03!^+ 2 +Jc,-«2e4.r»0 - 
-^ X J X J X 










o[ 






\ 




+ 






\ 










\v 


y 








l-M 


3-2- 














w 


\ 






















Lift Coefficient 


\\ 


\ 
























\^ 


\ 






















\ 


\ 






















\ 


V 


\ 




















.30 
.20 

.10 



C 


\\ 


\ 


\ 


\ 


\, 


















l\\ 


\ 


V 


\, 


\ 


\ 
















\\ 


\ 


\ 


s 


\ 




\ 


. 












\v 


\\ 


N 






^ 


^ 




"^ 


-^ 



-80 






,v 


N 


V 


^, 


"^ 


^ 






-^ 


-.^ 



^t4 




v 


C:: 


^ 


"^ 


^~-~ 


^ 






^ 


— 




V 








^^ 


^ 


mn 















1 1 \ 1 1 \ ' 

(246 

, 1 Wetted jjtngth- BeomR 


btio. 


'x 


) 

1 



Schematic for Heave 
Calibration Dimensions 

Fig.-§- 



Undistorbed 
Water Surfnce 



Modsl 




ActaaL Water Surface 



Lift of a Two- Dimensional Flat 
Plate as a Function of Froude Number 



Fig. 



































/5 






























































in 


1 






























i u 
































I 


4 




























IT 


-\ 










^^ 




















1 1 


V 




/ 


^ 




















u 


1/ 2 5 4 5 S 7 






t 


^E 






























I 





























Fr=;^ 




-- CO 



(£> 



^ i 

o 
oc 

E 
o 

CD 



0» 

s 

-J 

-o 
ro 0) 



CVJ 



o 
cvi 



to CJ ® ^ 

0(40^* |D|u»ui|i«dx«~lQ / p*4nduioo~lQ 



o 
o 









^> ° 


o 
o 


o 
o 








a 






o 












t 






o 




O 


• 






a ° 




o 


*w 












• 












O. 












O 






\ 




o 

8 




Ki 








o 








^ 




T> 


c 
o 


(D 


<b 




Oo 


• 












9 

a. 
E 

o 


« 


o 


> 

• 




oo 

Oo 
o 


o 


o 










-1 


A 








o 


O 


•E 




a 




^ o 


O 


"^ 




< 







o 










o 


^ 












o 






"»o 


o 


* 










o 

oo 


o 
o 





CP 



oo 



o 

00 

o 

•^ o o 

ex ^ O 

3 CM lO 



o <j a 



(0 



o 

in '^ 

o 
q: 

E 
o 

CD 



c 
a> 

-J 



5 



-CVJ 



+ 



O 

cvi 



^ oj 00 ^ 

Oi;D^ *|0*u»ui|J9dxt "Iq / p9|ndujoolQ 





,70 
.60 












L . 


















Empirically Corrected Values of 
C|_ vs. X 

Fig. 10 


































































i\ 


























\ 
























o 


J 


\\ 


V 






















Coefficient 

:> c 


\ 


\ 






















\\ 


\ 


v 




















'3 


.-rvy 


A 


\ 


N 


X 




















.30 
.20 
.10 


\\ 




\. 






-^ 
















\ 


\ 




"^ 








^^ 




. 30' 






iV 




V 


^^ 




^ 




-^ 




. 24« 






\\ 


k 








^ 












^ 


V 


^ 


^ 
















v^ 


K 


^^ 


■~~~- 









■ 1 




- !£• 


9- 














V 


^ 


^'^*— 







— 
































. 




C 


H 2 ' 4 ' 4 ' ^ ' I'O 
1 1 1 Wetted Length - Beam Rotio, A | | 





. .o 



o 



o 
O 




T3 O 
<D ■•- 

IS 

E.i 
<§» 
^ i2 



O 
(A 








o 

Q. 

E 
o 
O 



'!-' 







BOLLAY 

WeiNIG 

O EXPERIMENT 

a • ANGLE OF ATTACK 

Cm ' NORMAL FORCE 
" COEFFICIENT 



COMPARISON of WEINI6 ond BOLLAY AIRFOIL THEORIES 
for the EFFECT of LOW ASPECT RATIO 

Flg.J2_ 



LIFT COEFFICIENT vs. ANGLE OF ATTACK 
for FLAT PLATE AIRFOILS 

Fig. '3 



1.2 


- 


X = 7. 46 


1.0 




/ 
/ 


.8 




/ O 

/ 

/ o 


L .6 




/ o 


.4 




/o / 

/O / 


.2 


" 









.^->^l 1 1 1 


t 


r t t 1 r 




10 20 30 40 




Theory with 

rh bated on 

a 

Theory with 

rh bosed on 
C^e 
O Experiment 

V J L 

A = b» - b 

a ~ Angle of 

Attack 



10 20 30 40 



Figures for APPENDIX H 
Fig.J4_ 










.> 




1 1 






^ 


— - 










' — 












/OT^J 










y^ 


I 


















U-io 




K«! 


/ 






















^0.50 

Pi 




/ 


























/ 


f 




1 


L 


















o 9/r 




/ 






1 


^h 


















u»4>D 




/ 


























o 


^ 


/ 






\ 


r 


















v^ 

C 


) 


1 




< 


) 

,* 


3 


^ 


- 


> 


5" 


1 


f 
7 




7 



Pn - ::^;^ fi^^ »^^ 



/iT 


















C^ 10 

OL 


















I 


















V. 














— 


r\ 


-^^ 












i 


/ £. 3 4 5" 6 7 




\ 
1 


X 















Fig. 14 (cont'd) 

cm 




rrs i3S 



(b) 

























/_ 












,-/r 













/ 
















/ ^ 3 4 5" <^ ^ 



r-r' ■r-' 




K 



0.(6 

a2 



"^-^ 






















-^ — -__ 
















Fic^. 130 



fO 



^^^' 

od 



30 




.n\ 



Farshiug ^^^^eien" 
The 1-^T^ co..iiJ- 
flat planing surf.ee. 



TtiesiS' 
F236 



^3031 
Farshing 

Thif lift coefficient of flat 
planing surfaces.